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EXPLORATIONS OF THE ERDOS-FALCONER DISTANCEPROBLEM AND RELATED APPLICATIONS
A Dissertation
presented to
the Faculty of the Graduate School
University of Missouri
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
by
STEVEN SENGER
Dr. Alex Iosevich, Dissertation Supervisor
April 2011
c©Copyright by Steven Senger 2011All Rights Reserved
The undersigned, appointed by the Dean of the Graduate School, have examined thedissertation entitled
EXPLORATIONS OF THE ERDOS-FALCONER DISTANCE PROBLEM ANDRELATED APPLICATIONS
presented by Steven Senger,
a candidate for the degree of Doctor of Philosophy and hereby certify that in theiropinion it is worthy of acceptance.
Professor Alex Iosevich
Professor Peter Casazza
Professor William Banks
Professor Jan Segert
Professor Sergei Kopeikin
ACKNOWLEDGMENTS
First and foremost, I wish to humbly offer my thanks to Professor Alex Iosevich, my
advisor, teacher, and friend. He has taught me carefully, thoughtfully, and with a
limitless amount of patience. I would also like to express my gratitude to Professor
Pete Casazza for his guidance and collaboration. I am very grateful for my interactions
with Professor Bill Banks, Professor Jan Segert, and Professor Sergei Kopeikin, who
have always been willing to offer help and suggestions over the last decade.
I also want to thank, in no particular order, Nancy Brown, Laura Poe, Kyle
Gustafson, Professor Mel George, Professor Ignacio Uriarte-Tuero, Professor Kevin
O’Bryant, Professor Mel Nathanson, Professor Jason Aubrey, Nets Katz, and Wei He
for their kindness and advice through different times in graduate school.
Very special thanks go to my parents, Phil and Sharon Senger, as well as my
sister, Susan Mason, for their unending support. I am extremely blessed to have such
a kind and loving family.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
QUOTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viii
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
2. Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1. Background and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1. Discrete setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2. Continuous setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.3. Finite field setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.4. Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2. Point configuration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2. Applied results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3. Hinge estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2. Quantifying homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3. Well-distributed settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
3.4. Sharpness examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5. Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.1. Proof of Lemma 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6. Relationship to the continuous setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
iii
3.6.1. Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6.2. Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
4. Bounds on d-simplices in finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2. Proof of the first result (Theorem 4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3. Proof of the second result (Theorem 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5. Dot Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2. The n12 argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
5.3. The n23 argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36
5.4. Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.5. Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6. Systems of orthogonal vectors in finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2. Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3. Sharpness examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7. Sharpness examples related to the unit distance problem . . . . . . . . . . . . . . . 58
7.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.2. Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.2.1. Combinatorial underpinnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.2.2. Measure construction using combinatorial information . . . . . . . . 64
7.3. Matilla’s construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.4. Inequality (7.2) and the discrete incidence theorem . . . . . . . . . . . . . . . . 70
iv
7.5. Valtr’s example applied to vector spaces over Zq . . . . . . . . . . . . . . . . . . . 72
8. Applied results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.2. Semi-circular frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.3. Check matrices with entries from a finite field . . . . . . . . . . . . . . . . . . . . . 81
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
v
LIST OF FIGURES
Figures Page
1. Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2. Graph construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3. Well-distributed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4. r-iterated well-distributed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5. Valtr construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6. Continuous construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7. Error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8. Corner error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
vi
EXPLORATIONS OF THE ERDOS-FALCONER DISTANCE PROBLEM AND
RELATED APPLICATIONS
Steven Senger
Dr. Alex Iosevich, Dissertation Supervisor
ABSTRACT
We study a collection of geometric combinatorial problems which are closely re-
lated to the classical Erdos-Falconer distance problem. This problem explores the
relationship between the size of a subset of a vector space, and the size of the set of
distances determined by pairs of its points. We explore a number of generalizations
of this problem in various settings. In particular, we offer non-trivial bounds on the
number of times a pair of distances can occur in triples of points in subsets of the
plane, as well as bounds on simplices determined by subsets of vector spaces over
finite fields. We also give bounds for related quantities, such as dot products and
non-Euclidean distances, in several different settings. Also, we include applications
and related results which are useful in frame theory and coding theory.
vii
QUOTE
“Math is hard. Let’s go shopping!”
–Barbie
viii
Chapter 1
Introduction
There are many questions in discrete geometry which ask for what kinds structures
or configurations are to be guaranteed or prohibited in a set of points in some vector
space. We will consider subsets of Rd which are both finite and infinite, as well as
subsets of vector spaces over finite fields. This will be followed with some applications
and related work. To begin, we introduce what could be considered as the most
elementary specimens of these problems, which we collectively refer to as the Erdos-
Falconer distance problem.
One of the most elementary objects to study is the distance between points. Given
a P ⊂ Rd, we define the distance set of P to be
∆(P ) = {|x− y| : x, y ∈ P} ,
where |x| is the standard Euclidean norm.
Two of the most elementary questions involving distances are the Erdos distance
problem, introduced in [16], and the unit distance problem. The former asks for the
number of distinct distances which are guaranteed to be determined by a finite point
set in Rd, while the latter asks how often a pair of points can be separated by a single
fixed distance in a finite point set in Rd.
1
In [17], Falconer posed a related question of subsets of Rd which need not be point
sets. However, in this setting, we need different notions of size. These and related
problems have been studied, using analogous techniques, in vector spaces over finite
fields. Of course, the standard Euclidean norm may no longer be well defined. This
requires a slightly different notion of distance.
There are two important ways to generalize the notion of distance as we use it
here. The first is to investigate configurations of several points, such as triangles, or
other simplices. These can be characterized by the distances between pairs of points
in small collection. We will offer bounds on how often pairs of distances occur when
measured from a single point in R2, as well as bounds on d-simplices in a d-dimensional
vector space over a finite field.
The second is to consider objects which are different from the Euclidean norm.
In order to get a better understanding for what is inherent in problems like those
mentioned above, it is useful to study other quantities determined by point pairs. We
will consider dot products in R2, non-Euclidean metrics in several settings. We also
present bounds to guarantee systems of orthogonal vectors in vector spaces over finite
fields. In some cases, these explorations have merely exposed weaknesses in earlier
approaches to the traditional problems, while others have had applications in their
own right.
Finally, we discuss some consequences of these explorations. We present related
results in frame theory and coding theory.
2
Chapter 2
Overview
2.1 Background and notation
In this section, we discuss the three main distance problem settings: discrete, contin-
uous, and finite fields, and some important generalizations thereof.
2.1.1 Discrete setting
In order to clarify the exposition, we introduce some asymptotic notation. In what
follows, if two quantities, X and Y , vary with respect to some parameter, n, we say
X . Y if X ≤ CY , for some constant, C > 0, which does not depend on n. We write
X ≈ Y when X . Y and Y . X. Moreover, let X / Y mean that X ≤ CεnεY ,
for any ε > 0, with Cε > 0 depending on ε, but not n. We often use the latter
notion when trying to bury logarithmic dependence on a parameter n. Also, we will
occasionally use (and possibly reuse) c to stand for an unspecified constant. We freely
use both #E and |E| to denote the appropriate size of a set E. Typically we use #E
to emphasize that the set is discrete.
The conjecture for the Erdos distance problem is that, given a set, E, of n points
3
in Rd,
#∆(E) ' n for d = 2, and
#∆(E) & n2d for d ≥ 3.
Guth and Katz have produced the desired bound for the Erdos distance problem
in the plane, in [21]. In higher dimensions, Solymosi and Vu hold the best result,
from [46].
The conjecture for the unit distance problem in the plane is that no distance can
occur more than / n times in a set of n points. Given a set of n points, Spencer,
Szemeredi, and Trotter, in [47], showed that a unit distance can occur no more than
. n43 times. The proof is based on a modified version of the celebrated Szemeredi-
Trotter point-line incidence theorem below, from [50].
Theorem 2.1. Consider any a set of n points and m lines in the plane. Then we
are guaranteed that I, the number of pairs, (p, l) such that p is one of the n points,
and it is incident to l, one of the m lines, is bounded above by
I . (nm)23 + n+m.
A construction due to Lenz, [8], answers the second question in dimensions four
and higher. We address Lenz’s construction in Chapter 7. See [8] for more information
on these, and related problems. These questions shall be referred to as the discrete
setting.
2.1.2 Continuous setting
In order to approach infinite subsets of Rd, we will need some different ways to
quantify size. To this end, we recall the following two definitions.
4
Definition 2.2. Given a set E ⊂ Rd, let UE,δ denote the set of all finite covers of E
whose members have diameter ≤ δ. Define the s-dimensional Hausdorff measure to
be
Hs(E) = limδ→0
infUE,δ
∑Uj∈UE,δ
(diam(Uj))s
.
Definition 2.3. Given a set E ⊂ Rd, define the Hausdorff dimension to be
dimH(E) = sups{Hs(E) <∞} = inf
s{Hs(E) > 0} .
With this in tow, we can state the Falconer distance problem as, “Find the smallest
number, s, for which any subset, E ⊂ [0, 1]d, with Hausdorff dimension greater than
s, is guaranteed to satisfy the inequality (2.1) below.”
L1 (∆(E)) > 0, (2.1)
where Lk denotes the k-dimensional Lebesgue measure. The key estimate used to
find sufficient conditions for (2.1) to hold in the initial paper was
(µ× µ) ({(x, y) : t ≤ |x− y| ≤ t+ ε}) . ε, (2.2)
for an appropriate Borel probability measure µ, a given distance t, and ε > 0.
It is conjectured that any subset of Rd with Hausdorff dimension strictly greater
than d2
will have a distance set of positive Lebesgue measure. In his initial offering on
the problem, [17], Falconer showed that s > d+12
was sufficient to guarantee a distance
set of positive Lebesgue measure. This was improved to s > 43
in the plane by Wolff
in [57], and later in higher dimensions to s > d2
+ 13
by Erdogan in [14].
We shall refer to problems like this one introduced by Falconer, as continuous prob-
lems. Even superficially, one can see the resemblance between conditions to guarantee
5
(2.1) and the Erdos distance problem, and the similarity of sufficient conditions for
(2.2) and the unit distance problem.
2.1.3 Finite field setting
Let Fq denote the finite field with q elements, and Fdq will be the d-dimensional vector
space over this field. Since there is no Euclidean metric in this setting, we have to
work with a closely related object, which, by abuse of language, we will also refer to
as a distance. Let the x and y be two points in Fdq , with coordinates (x1, x2, . . . , xd)
and (y1, y2, . . . , yd), respectively. We will define the distance between them to be
‖x− y‖ =d∑j=0
(xj − yj)2.
One of the many attractive features of this definition of distance is that it is invariant
under group actions by the associated orthogonal group with entries in Fq.
So, given a subset, E ⊂ Fdq , how big does E have to be in order to guarantee that
distance set has about as many elements as possible? To be precise, if |E| & qs, for
which values of s can we say that
#∆(E) & q,
where here, the ∆(E) refers to the distances as defined for Fdq . The conjecture
is that, for even dimensions, #E & qd2 will guarantee #∆(E) & q, while in odd
dimensions, one needs that #E & qd+12 in order to guarantee the large distance set.
The finite field setting is quite interesting when analyzed along with the previous
two, as it has aspects of both, without completely taking on the characteristics of
either. This problem was introduced by Bourgain, Katz, and Tao in 2004 in [7]. In
6
2007, Iosevich and Rudnev, in [30], obtained bounds with explicit exponents, which
are sharp for odd dimensions. Analytic number theory played a key role in these
results.
2.1.4 Generalizations
We begin with a helpful definition.
Definition 2.4. Given a subset E ⊂ Rd, define the upper Lebesgue density, or some-
times just upper density as
d(E) = lim supn→∞
|E ∩Bn(−→0 )|
|Bn(−→0 )|
,
where Br(x) denotes the ball of radius r centered at x,−→0 denotes the origin.
Furstenberg, Katznelson, and Weiss, in [18], showed that a δ-neighborhood of
any subset of the plane with positive upper Lebesgue density contains the vertices of
every sufficiently large triangle. Later, Bourgain generalized this to higher dimensions,
while demonstrating the necessity of the δ-neighborhood with arithmetic progressions,
which are examples of degenerate simplices, in [5].
In [19], many families of non-Euclidean distances are studied, using clever modifi-
cations of the canonical Erdos distance problem techniques. Similarly, for the study
of the unit distance problem, Matousek showed, in [43], that most norms admit no
more than n1+ε unit distances for any set of n points in the plane, and for any ε > 0.
This technique, however, could not be applied directly to the Euclidean norm.
7
2.2 Summary of results
Many of the results presented here are for generalizations of the point configura-
tion problems along the lines of those described in the last section. However, these
explorations have brought up a number of results in related areas.
2.2.1 Point configuration results
In the discrete and continuous settings, we offer bounds on the number of times a
pair of distances can occur between a single point and two other points in a large
class of subsets of Rd. To quantify this, we present the following definition.
Definition 2.5. Given two real numbers, α and β, we define an (α, β)-hinge to be a
triple of points (x, y, z) ∈ Rd × Rd × Rd such that
|x− y| = α and |x− z| = β.
In general, the term hinge applies to any pair of distances determined by three points.
In particular, we offer bounds, in the discrete and continuous settings, for how
often any fixed hinge can occur in subsets of the plane. Both results offer bounds
which are worse than those given in [10], but the discrete bound holds for a wider
class of sets. We offer examples of sets for which we can give bounds where the latter
bounds do not hold.
In the finite field setting we offer a lemma which proves crucial to prove a theorem
on the number of distinct simplices present in a subset of Fdq . We also present a simple
argument which extends the range of the previous result, but relies on a separate and
highly nontrivial theorem.
8
We also offer bounds on the number of distinct dot products which are guaranteed
to be present in any large finite set of points in the plane, as well as improved bounds
for certain classes of sets. In the finite field setting, we give sufficient conditions which
guarantee the existence of a system of k orthogonal vectors in Fdq . Of course, k ≤ d.
Finally, we give a family of counterexamples to many incidence problems related
to the unit distance problem, for several types of metrics. In particular, we extend
an example given by Valtr, [56], for the discrete setting, to the continuous and finite
field settings. We also investigate consequences of an example given by Mattila in
[39], which defy previously held intuitions on the unit distance problem.
2.2.2 Applied results
Frame theory is a wide subject, which was arguably introduced in [13]. One way of
viewing a frame is as a collection of vectors which form an over-complete basis for
some Hilbert space. That is, there are often many more vectors present in the frame
than needed to span the space. This underlying space may have a finite or an infinite
number of dimensions. Frames provide redundant information, which is often quite
useful. See [12] for more details.
The first result we prove involves error bounds for a simple error-correcting scheme
used with a particular class of frames called semi-circular frames. These are frames
whose vectors are evenly distributed along unit semicircles centered at the origin.
Such frames have desirable qualities which can be quantified using techniques related
to those often employed to study point configurations.
The theory of codes for error correction involves carefully adding redundant in-
formation to increase robustness against data corruption. Objects which vary in
9
complexity from the simple checksum bit to an entire spoken language can be viewed
as codes. See [22] for more information on coding theory.
One of the key features of many codes is called a check matrix, which have entries
from a finite field. The second applied result is a corollary of the bounds on systems
of orthogonal vectors in vector spaces over finite fields mentioned above. If one has
only a subset of entries in a vector space available for use, the bounds we offer here
can be used to discern whether or not a given type of code can be generated from the
available subset.
10
Chapter 3
Discrete and continuous hingeestimates
3.1 Overview
Given a set E ⊂ Rd, points x, y, z ∈ E, and two fixed distances, α and β, we define the
triple (x, y, z) to be a (α, β)-hinge if |x− y| = α and |x− z| = β. Let Λα,β(E) denote
the set of (α, β)-hinges in a set E, and #Λα,β(E) denote the number of elements in
that set. While the study of hinges has intrinsic interest on its own, estimates on the
number of hinges determined by a set in R2 can often be employed to gain estimates
on the number of triangles, whose vertices are in the set. Higher dimensional analogs
of hinges can then be used to talk about higher dimensional simplices.
3.2 Quantifying homogeneity
In much of what follows, we will rely heavily on the distribution of points in order to
apply certain techniques. Not all point sets are eligible for some of these techniques,
so we need to quantify how well-distributed a point set is. The perfect tool for this
is called s-adaptability, which we now define and discuss.
Definition 3.1. We say that a set P of n points in [0, 1]d is s-adaptable if it satisfies
11
the following two conditions:
i)1(n2
) ∑x 6=y∈P
1
|x− y|s. 1. (3.1)
ii) min{|x− y| : x 6= y ∈ P} ≥ n−1s . (3.2)
Notice, by definition, that if a set in [0, 1]d is s0-adaptable, for some s0 ≤ d, then
it is also s′-adaptable for any s′ < s0. Given a set P , of points, let S(P ) denote the
highest value of s for which E is s-adaptable. To extend the idea of s-adaptability to
subsets which are not contained in the unit cube, simply uniformly scale them down
until they fit, and check the definition. There are sets, even in the plane, which are
not s-adaptable for pertinent ranges of s. We will discuss this further in Subsection
3.6.2.
The notion of s-adaptability is useful for translating between discrete and contin-
uous settings freely. Let Br(p) denote the ball of radius r centered at p, and χBr(p)(x)
denote the indicator function of such a ball. Now, for a set P of n points in Rd, let
ε = n−1s , and define a measure
dµsP =1
nnds
∑x∈P
χBε(p)(x)dx.
If P has no more than one of point in any ball of radius ε, then we are guaranteed
bounds, above and below, of the s-dimensional energy integral,
Is(µsP ) =
∫ ∫|x− y|−sdµsP (x)dµsP (y) ≈ 1.
Let P denote the support of the measure µsP . Notice that P consists of ε-balls
centered at the points of P . If many points in P were closer than ε, we will lose mass,
12
and some characteristics from the point set might not be preserved in the transformed
set, invalidating our analysis. The reason is, Hausdorff dimension captures more than
just size; it also measures the distribution of mass in a given set. We will revisit this
discussion in Section 3.6. For more information on s-adaptability and the conversion
mechanism, see [26], [28], and [31]
3.3 Main result
With all of this in tow, we present the main results.
Theorem 3.2. Let P be an n−1s -separated set of n points in [0, 1]2, with 2 ≥ s > 3
2,
and α, β > 0, then
Λα,β(P ) / n43
+ 1s .
For s ≤ 32, the result of Theorem 3.2 is trivial, as we will show in the next subsec-
tion. Also, in the plane, s ≤ 2, by definition. Let Λ(P ) denote the set of all distinct
hinges determined by points in P . That is,
Λ(P ) = ({(|x− y|, |x− z|) : x, y, z ∈ E})
Although in [10], there is a stronger version, we include the following theorem to
illustrate the mechanism of translating a discrete result to a continuous result, and
to see how the converted results compare. This theorem requires another definition.
Definition 3.3. A subset E ⊂ Rd, with Hausdorff dimension s, will be called Ahlfors-
David regular if there exists a probability measure, µ, supported on E, and constants,
Cupper and Clower, such that for every x ∈ E,
Clowerrs ≤ µ(Br(x)) ≤ Cupperr
s. (3.1)
13
Theorem 3.4. Given an Ahlfors-David regular set E ⊂ [0, 1]2, with Hausdorff di-
mension s > 95,
L2(Λ(E)) > 0.
3.4 Sharpness examples
Here, we will show that the fact that the points are s-adaptable, with s bounded
below, is key to the proof of Theorem 3.2. First, we prove the trivial upper bound on
the number of hinges in general point sets. Next, we show that this bound is sharp.
Then, we illustrate how that the s-adaptability condition guarantees an improvement
over these bounds. Finally, we indicate why the restriction s > 32
should be expected
in hinge problems.
One of the simplest hinges to consider is the (1, 1)-hinge. Note that any three
points which form a (1, 1)-hinge also form, by definition, an isosceles triangle, with
at least two sides of unit length. First, we present the trivial bound for any hinge.
What follows is a bound on the number of (1, 1)-hinges in any set of points in the
plane. Notice that every such hinge is also an isosceles triangle with unit repeated
length, so we present this as a bound on the number of isosceles triangles of repeated
length one.
Theorem 3.5. Given a set, E, of n points in the plane, there can be no more than
n2 distinct unit isosceles triangles determined by the points in E.
Proof. Given any pair of points, draw unit circles centered at each point. These
two circles can intersect in no more than two places. So, when one of the points of
intersection is a point of E, there is a unit isosceles triangle described by this point
14
and the original pair of points. Note that such a configuration is necessary for an
isosceles triangle to be determined by the points in E. So there can be no more than(n2
)≈ n2 distinct unit isosceles triangles.
Proposition 3.6. There exists a set E of n points in the plane such that there are
& n2 distinct unit isosceles triangles whose vertices are determined by points in the
set E.
Proof. We shall construct a set which determines & n2 distinct unit isosceles triangles.
First, let the origin be in E. Draw the unit circle around it, and call it C. Let p1
denote the point with coordinates (1,0). Notice that it lies on C. Draw the unit circle
centered at p1, and select some point p2 in the upper half-plane such that it does not
lie on the intersections C and the unit circle centered at p1. Now draw a unit circle
around p2, and draw circles of radius |p1 − p2| around both p1 and p2. Now draw
p3 on C such that it is not on any of the intersections of any of the other circles.
Repeat this process until n points are on C. We can continue indefinitely, as we are
prohibiting only a finite number of points on C from being in E, but there will always
be infinitely many more from which to choose.
We can modify the above construction to obtain possible sharpness examples for
s-adaptable sets. Construct a set Ps of n points in the unit square as above, but
instead of putting n points on the unit circle centered at p above, we can only put n1s
points on it, by the s-adaptability condition. Reasoning as in the previous paragraph
yields the following proposition.
Proposition 3.7. There exists an s-adaptable set of points, P ⊂ R2, which deter-
mines & n2s distinct unit isosceles triangles.
15
We include the following corollary to exhibit the consequences of s-adaptability.
Corollary 3.8. Given a set, P , of n points in the plane, such that there are no
more than c1 points in each closed cell of the n12 × n 1
2 lattice, the number of isosceles
triangles, with repeated side length c2n12 , for given constants c1 and c2, is / n
116 .
We get this by applying Theorem 3.2 with s = 2, and α = β = c2n12 . One should
note that this result is not implied by upper bounds on the number of unit equilateral
triangles determined by a point set.
Now we present a related continuous sharpness example introduced by Mattila,
in [39]. This will show that s > 32
is a natural lower bound for where we expect to
get results. Let Cλ ⊂ [0, 1] denote the Cantor set with parameter λ, constructed as
follows. Begin with the unit interval, and remove the middle λ-proportion. Remove
the middle λ-proportion of each of the two remaining intervals. Repeat this process
indefinitely, and you will have a subset of the unit interval with Hausdorff dimension
(log(2)/ log(1/λ)). Given a set I ⊂ [0, 1], let I − 1 denote the set of arithmetic shifts
of I, namely, {x − I : x ∈ I}. Define F to be (Cλ(γ) ∪ (Cλ(γ) − 1)), where λ(γ) is
the value of λ for which Cλ has Hausdorff dimension γ. Let Mγ = [0, 1]× F, and let
µ = Hγ|F × L|[0, 1]. Notice that Mγ has Hausdorff dimension 1 + γ, where Hη is
the η-dimensional Hausdorff measure. Given any ε > 0, elementary geometry reveals
that, given a fixed x0 ∈Mγ,
µ{y ∈Mγ : 1 ≤ |x0 − y| ≤ 1 + ε} & ε12
+γ,
which implies that
µ{(y, z) ∈Mγ ×Mγ : 1 ≤ |x0 − y|, |x0 − z| ≤ 1 + ε} & ε2(12
+γ),
16
which gives us
µ{(x, y, z) ∈Mγ ×Mγ ×Mγ : 1 ≤ |x0 − y|, |x0 − z| ≤ 1 + ε} & ε1+2γ.
If we choose a γ < 12, we will see that we cannot, in general, hope for the hinge
version of (2.2) to hold. This will be described in more detail in Section 3.6. For
more information, see [32].
3.5 Proof of Theorem 3.2
Proof. Without loss of generality, suppose α ≥ β. Now, if either α or β is much too
large or much too small, we will trivially get even better bounds, so we only examine
the worst possible cases, where α and β are between, say 1n
and 2. Let O denote the
set of circles of radius α or β, which are centered at points in P . First, we will get
upper and lower bounds on the number of incidences of circles of O and points in E.
When we compare these bounds, and use a little linear programming, we will get a
non-trivial estimate on the number of (α, β)-hinges.
Let I denote the number of incidences of points in P and circles in O. Since there
are n points in P , and two circle centered at each point, there are 2n such circles to be
considered. We now present a modified version of Theorem 2.1, the Szemeredi-Trotter
incidence theorem.
Lemma 3.9. Given n points, n circles of radius r1, and n circles of radius r2, centered
at each of the points, the number of incidences is bounded above by a constant times
n43 .
Applying Lemma 3.9, we see that
17
I . n43 .
First, we break the unit square up into n1s × n 1
s subsquares, each of side-length
n−1s . By the separation condition, there can be no more than a constant number of
points in each subsquare. This means that no circle can be incident to more than a
constant times n1s points. Let Oα(p) denote the set of points in P incident to the
circle of radius α centered at p. Let iα(p) = # (Oα(p)). Define Oβ(p) and iβ(p)
similarly.
Pαj = {p ∈ P : 2j ≤ iα(p) < 2j+1},
and
P βk = {p ∈ P : 2k ≤ iβ(p) < 2k+1}.
Now, we can count the number of incidences by summing, and break up the sum
dyadically
I =∑p∈P
(iα(p) + iβ(p)) ≈logn
1s∑
j=0
logn1s∑
k=0
∑p∈Pαj ∩P
βk
(2j + 2k
)' |Pα
l |2l + |P βm|2m,
We only needed to sum to log n1s , because no circle from O can contain more than
about n1s points. Now, for any pair of l and m, and any ε > 0, comparing these
bounds on I yields
|Pαl |2l + |P β
m|2m / n43 ,
which implies
|Pαl |2l + |P β
m|2m . n43
+ε. (3.1)
18
Now we will use some simple linear programming to get a bound on the number
of (α, β)-hinges. Notice that a given p ∈ P can contribute no more than i1(p)i2(p)
hinges which are (α, β)-hinges, where p is the vertex of the two legs. This is because,
there will be no more than i1(p)i2(p) choices of pairs of points, with one point from
the pair on each circle. Define Pl,m to be Pαl ∩ P β
m. Notice that points in Pl,m can
give no more than |Pl,m|2l+m (α, β)-hinges.
So we can be assured that
|Pl,m|2l + |Pl,m|2m . n43
+ε (3.2)
Given any pair of values of l and m, set a = logn |Pl,m|, b = logn(2l), and c =
logn(2m). So, irrespective of lower order terms, the bounds we have translated to the
following inequalities in a, b, and c:
a ≤ 1
b ≤ 1
s
c ≤ 1
s
a+ b ≤ 4
3+ ε
a+ c ≤ 4
3+ ε
If we maximize the objective function a+ b+ c with respect to these constraints,
we find a maximum of 43
+ 2s
+ 2ε. If we reverse the logarithms, we get |Pl,m|2l+m .
n43
+ 2s
+2ε. Since the points in the set Pl,m can give no more than |Pl,m|2l+m (α, β)-
hinges, we have the desired result.
19
3.5.1 Proof of Lemma 3.9
We prove Lemma 3.9 using techniques of Szekely, from [49]. Draw a graph with the
n points as vertices, and the arcs of the circles connecting adjacent points as edges.
Notice that the number of times two circles intersect each other can be no more than
2(2n)2. This is because there are(
2n2
)pairs of circles, and each pair can intersect
only twice. Observe that no pair of points can have more than eight edges connecting
them, as the centers of the circles would have to lie on the bisector of the two points,
and be one of two distances away from both points. So, there are no more than
four such circles incident to each pair of points, and each circle can contribute no
more than two edges to each point. Also, notice that any time there is a point-circle
incidence, we have to draw another unique edge. So the number of incidences is equal
to the number of edges. Recall the crossing number lemma in [49].
Lemma 3.10. Given a topological multi-graph, G, with v vertices, e edges, and a
maximum edge multiplicity of m, if e > 5mv,
cr(G) &e3
mv2,
where cr(G) denotes the crossing number of G, which is the maximum number times
edges of G must cross at some point which is not a vertex, for any redrawing of G.
For our setup, either I = e < 5mv ≤ 8n or
I3
8n2.
e3
mv2. cr(G) . n2.
In either case, the claimed estimate holds.
20
3.6 Relationship to the continuous setting
Here, we discuss some aspects of the general relationship between discrete and con-
tinuous results, as well as their relative behaviour in the context of hinges. First, we
will prove Theorem 3.4, to illustrate the conversion of discrete results to continuous
results. Then, we introduce the analogous result in [10]. Finally, we show how their
result can be converted into a discrete result, and compare.
3.6.1 Proof of Theorem 3.4
Given an Ahlfors-David set E ⊂ [0, 1]2, with Hausdorff dimension 32≤ s ≤ 2,
there exists a probability measure, µ, which satisfies (3.1). We want to show that
L2(Λ(E)) > 0. So we will prove that for any cover of Λ(E) by squares, Qj of radius εj,
the (µ×µ×µ)-measure of triples of points in E×E×E which contribute to a given
square, Qj, is small. In what follows, we consider an arbitrary Qj from any cover of
Λ(E). This Qj will be centered at the point (αj, βj), have Lebesgue measure ε2j , and
will consist of (α′, β′)-hinges, where αj− εj ≤ α′ ≤ αj + εj, and βj− εj ≤ β′ ≤ βj + εj.
To simplify notation, we shall refer to εj by simply ε throughout the rest of the
proof. Decompose the unit square into ε by ε squares. Let E ′ denote the centers
of each square which is in the support of E. Since µ is a probability measure, (3.1)
guarantees that the number of points in E ′ is about ε−s. So, let n = ε−s. Given any
(α, β)-hinge which occurs in E ′, define
M(α, β) := (µ×µ×µ) ({(x, y, z) ∈ E × E × E : α ≤ |x− y| ≤ α + ε; β ≤ |x− z| ≤ β + ε}) .
Now, by definition, we have that
M(α, β) . ε3s#Λα,β(E) . ε3s(n43
+ 1s + n
53 ), (3.1)
21
by Theorem 3.2, which, by definition of n is
. ε53s−1.
Now, since the Lebesgue measure of Qj is ε2, we need to find conditions on s which
will guarantee
M(α, β) . ε2. (3.2)
Notice that for s > 95, (3.2) holds. So, by the definition of sets of Lebesgue measure
zero, we have shown that Λ(E) does not have measure zero, and we are done.
3.6.2 Comparison of results
In [10], the following is shown.
Theorem 3.11. Given an Ahlfors-David regular set E ⊂ R2, with Hausdorff dimen-
sion strictly greater than 53,
L2(Λ(E)) > 0.
Notice that Theorem 3.4 is implied by Theorem 3.11, as the latter holds for dimen-
sions greater than two, has a wider range on s, and has no Ahlfors-David regularity
assumption. However, if we convert Theorem 3.11 to a discrete result, this result does
not imply Theorem 3.2. We now present a discrete version of Theorem 3.11.
Theorem 3.12. Let P be an s-adaptable set of n points in R2, with 2 ≥ s > 53, and
α, β > 0, then
Λα,β(P ) / n95 .
We do not get this result from the statement of Theorem 3.11, but from its proof.
The key estimate turns out to be analogous to (3.1), and we rewrite it here using
22
the notation of the proof of Theorem 3.4. Given a set E, in [10], it is shown that for
appropriate α, β, and ε.
ε3s#Λα,β(E) . ε2, (3.3)
when s > 53. Notice that for any point set, P , with S(P ) < 5
3, Theorem 3.12 cannot
be applied, and gives no result, where Theorem 3.2 does. However, if S(P ) ≥ 53,
Theorem 3.12 provides a stronger result. Notice that the Theorem 3.2 gives worse
exponents as S(P ) decreases, where Theorem 3.12 either gives a result or does not.
The reason for this is in the way that each approach to the general problem uses
the s-adaptability condition. The discrete argument uses s-adaptability only to limit
the number of incidences between points in the set and a given circle. The continuous
results often rely on energy estimates, which are affected by how clustered the mass of
a given set is. Also, continuous results which come from equations like (3.3) guarantee
the existence of sets of objects which are somehow separated. In this case, Theorem
3.12 could be employed to show that a large number of distinct hinges are present in a
given set, as well as quantify how different these hinges are from one another. While
perhaps results like Theorem 3.2 could also be used to guarantee a very large number
of distinct hinges, the method of proof could not say anything about how different
our distinct hinges must be. This appears to be one of the great trade-offs in working
problems with either method. Discrete methods can detect arbitrarily fine differences
between two configurations of a given type, but they do not seem to appreciably
quantify these differences. Continuous methods can quantify such differences easily,
but this then limits the range of differences which they can discern.
Earlier, we mentioned that there exist sets which are not s-adaptable for pertinent
23
ranges of s, even in the plane. See [31] for more information. Here, we describe the
construction of a family of sets for which Theorem 3.2 holds, but Theorem 3.12 does
not. The basic idea is to make a set of points which is “lattice-like” on multiple
scales. This can be done so as to satisfy the separation condition of Theorem 3.2,
while violating the corresponding energy condition.
Let n be of the form k4, for some integer k > 106. Let 32< t < 5
3. Start with an
n14 × n 1
4 lattice, scaled down to fit into the unit square. Now, replace each of these
points by smaller n14 × n 1
4 lattices, scaled such that the minimum distance between
any two points is n1t . For appropriate n and t, call such a set Fn,t.
Proposition 3.13. Any set Fn,t, described above, is not t-adaptable.
Proof. Clearly Fn,t satisfies the separation condition of Definition 3.1, so we now show
that the corresponding energy condition is not satisfied.
It(Fn,t) = n−2∑
x 6=y∈Fn,t
|x− y|−t
≥ n−2n12
∑x 6=y∈Q
|x− y|−t,
where Q ⊂ Fn,t denotes the n14 ×n 1
4 lattice, scaled down so that the smallest distance
between any two points is exactly n1t . We are only examining the energy contributed
by pairs of points in the same lattice, so it is enough to estimate this energy for one
24
lattice, and multiply that by the number of such lattices, n12 . So
It(Fn,t) & n−32
∑x∈Q
logn14−
1t∑
logn−1t
#{y ∈ Q : 2j ≤ |x− y| ≤ 2j+1
} (2j)−t
& n−32
∑x∈Q
logn14−
1t∑
logn−1t
(2jn
1t
)2 (2j)−t
& n−32n
12n
2t
logn14−
1t∑
logn−1t
(2j)2−t
& n2t−1(n
14− 1t
)(2−t)≥ n
12− t
4 ,
which has a positive exponent, thus violating the energy condition of Definition 3.1.
We believe that the bound on Theorem 3.2 could be improved if we could find a
special version of the Szemeredi-Trotter incidence theorem which dealt with sets of
s-adaptable points. One reason to hope for this lies in the fact that the version of
sharpness example for the Szemeredi-Trotter point-line incidence theorem in the unit
square, a n13 × n 2
3 “lattice” of points, is s-adaptable only for s ≤ 32. However, it is
possible that such an estimate may not hold in general.
25
Chapter 4
Bounds on d-simplices in finitefields
4.1 Overview
As we mentioned in Subsection 2.1.4, Furstenberg, Katznelson, and Weiss, in [18], and
Bourgain, [5], studied conditions on subsets of Rd which determined many simplices.
More recently, Akos Magyar has exlpored similar problems in the integer lattice in
[37] and [38]. He showed that if a subset E ⊂ Zd has positive upper density, then all
sufficiently large dilates of any given k simplex can be found in E, for d > 2k + 4.
These methods do not work, however, when they are employed to study d-simplices.
In what follows, we offer improvements to an analog of this question in vector
spaces over finite fields. In [23], Hart and Iosevich show that if E ⊂ Fdq , with |E| &
qdkk+1
+ k2 , then E contains a copy of every simplex, up to translation and actions of
the orthogonal group, Od(Fq). However, the useful range of this result is limited to
d ≥(k+1
2
). Le An Vinh has also studied these and related problems in [53], [54], and
[55].
In this chapter, we will be dealing with only d-simplices. Let ∆k denote a k-
simplex, that is, a set of k + 1 points which span a k-dimensional subspace. Given
26
E ⊂ Fdq , let the set of k-simplices determined by E up to congruence be denoted by
Tk(E) = {∆k ∈ Ek+1} / ∼
where two k-simplices are equivalent if one is a rotated, shifted, reflected copy of the
other.
We will see that a given simplex can be completely determined, up to rotation,
shift, and reflection by the mutual distances between every pair of points. That is,
Tk(E) ⊂ F(k+12 )
q (see Lemma 4.3 below).
The main results of this chapter are Lemma 4.6 and Theorem 4.2. Lemma 4.6
proves to be an integral part of an estimate on the number of d-simplices determined
by subsets of Fdq . The first lemma is a technical result which finishes the proof of the
next stated theorem, as seen in [11].
Theorem 4.1. Let E ⊂ Fdq where Fq has characteristic > 2, |E| ≥ ρqd for q−1/2 �
ρ ≤ 1. Then, there exists c > 0 so that
|Td(E)| ≥ cρd−1q(d+12 ).
The second main result allows for a slight improvement of Theorem 4.1, which is
an extension of a result from [9]. The difference between this theorem and the last is
that there are fewer factors of ρ in the improved result.
Theorem 4.2. Let E ⊂ Fdq where Fq has characteristic > 2, |E| ≥ ρqd for q−12 �
ρ ≤ 1. Then, there exists c > 0 so that
|Td(E)| ≥ cρq(d+12 ).
27
Remark 1. The assumption that |E| ≥ ρqd implies that the number of (d+ 1)-point
configurations determined by E (up to congruence) is at least
|E|d+1
ρqd · q(d2)≥ ρdq(
d+12 ),
since the size of the subset of the translation group that maps points in E to a set
of size |E| is no larger than ρqd and the rotation group is of size ≈ q(d2). This result
shaves off a power of ρ from this trivial estimate.
4.2 Proof of the first result (Theorem 4.1)
We will prove Theorem 4.1 by using Theorem 4.4, from [11], to get an estimate on
the hinges determined by points in E. With this estimate in hand, we will be in one
of two cases. The first case will be handled by another result from [11], which will be
stated as a technical lemma. The second case will be handled by another technical
lemma, Lemma 4.6, which we will prove below.
Basically, the we reduce the problem to a statistical estimate about hinges. After
that, we use a pigeonholing argument to show that there is a point, x ∈ E, which is
the shared vertex of many copies of a particular type of hinge. Now, we are down to
the two cases. In the first case, the number of transformations which map the hinge
to itself is small, so we get the desired number of d-simplices by the first technical
lemma. If not, then we need to employ the second technical lemma which is the first
main result of this chapter.
We start with the statistical reduction. We observe that if |E| ≥ ρqd, for ρ as
above, then it suffices to show that this implies that
28
|{ai,j}1≤i≤j≤d+1 ⊂ Fq : |Ra(E)| > 0}| ≥ cρd−1q(d+12 ), (4.1)
where
Ra(E) = {(y1, . . . , yd+1) ∈ E × · · · × E : ‖yi − yj‖ = ai,j}.
This follows immediately from the following simple linear algebra lemma. The
proof of this lemma is given in [11].
Lemma 4.3. Let V be a simplex with vertices Vi ∈ Fdq, where i = 0, . . . , k. Let W be
another simplex with vertices Wi ∈ Fdq for i = 0, . . . , k. Suppose that
‖Vi − Vj‖ = ‖Wi −Wj‖ (4.2)
for all i, j. Then V ∼ W in the sense of Tk(E).
The main estimate, proved in [11], is the following:
Theorem 4.4. Suppose that αi ∈ Fq\{0} for i = 1, . . . , d are all fixed, and let E ⊂ Fdq.
Then,
|{(x, x1, . . . , xd) ∈ E × · · · × E : ‖x− xi‖ = αi}| =|E|d+1
qd(1 + o(1))
whenever |E| � qd−12 .
This implies that there exists x ∈ E so that
|{(x1, . . . , xd) ∈ E × · · · × E : ‖x− xi‖ = αi}| ≥|E|d
qd(1 + o(1)). (4.3)
29
Fix a d-tuple α = (αi)di=1, with αi ∈ Fq\{0}, for i = 1, . . . , d. Define a pinned hinge
set hx,α to be the set {(x1, . . . , xd) ∈ E×· · ·×E : ‖x−xi‖ = αi}. Let Mx,α ⊂ Od(Fq)
denote the set of orthogonal matrices which maps the set hx,α to itself. Now, let
Ai := {xi ∈ E : ‖x− xi‖}.
Now, we will present two lemmas. Each one deals with one of the two possible
cases. The first case is dealt with in the first lemma, which is proved in [11]. The
second case is handled in the second lemma, which we prove below.
Lemma 4.5. Given the above setting, if |Ai| ≤ ρqd−1 or |Mx,α| ≤ ρq(d2). Then, we
can find at least cρq(d2) distinct d-point configurations among the sets Ai.
Lemma 4.6. Given the above setting, if |Ai| > ρqd−1 and |Mx,α| > ρq(d2). Then, we
can find at least cρq(d2) distinct d-point configurations among the sets Ai.
Proof. It is worthwhile to point out that since Ai are indexed according to the dis-
tances αi, it is possible that Ai = Aj. Also, although the sets Ai are not themselves
spheres, they are subsets of spheres and therefore inherit some the intersection prop-
erties of spheres. Recall that we are working with the hinge hx,α = {(x1, . . . , xd) ∈
E × · · · × E : ‖x− xi‖ = αi}.
We start by picking a point of the set A1, call it a1. We want to know how many
distinct distances occur between a1 and points in the set A2. To answer this, we
count how often a given distance occurs between a1 and points on A2. This amounts
to intersecting E with two spheres: one sphere of a given radius, centered at a1, and
the set A2, which is a sphere intersected with E. The intersection must contain fewer
than qd−2 possible points on the set A2 which are a given distance from a1. Since
|A2| > ρqd−1, there must be at least ρqd−1/qd−2 = ρq different distances between a1
30
and points on A2, by pigeonholing.
For each of the ρq choices of a2 which are different distances from a1, we need to
find the number of 3-point configurations that a1 and a2 can make with points on
A3. Now we are intersecting E with spheres of two (possibly the same) radii about
a1 and a2 with the sphere containing S3. There can be no more than qd−3 points
in this intersection, which would each correspond to the same 3-point configuration.
So there must be ρqd−1/qd−3 = ρq2 distinct 3-point configurations for each of the ρq
different pairs we found before, which gives us a total of ρq · ρq2 = ρ2q3 different 3-
point configurations. Repeating this process, we see that we will pick up ρqp different
(p− 1)-point configurations at each step. If we multiply all of these together, we will
get a grand total of
ρq · ρq2 · · · · · ρqd−1 = ρd−1q(d2) (4.4)
distinct d-point configurations.
From Lemma 4.5 and Lemma 4.6, we see that in any case, there exist no less than
cρd−1q(d2) many distinct d-point configurations. Since this holds for any fixed vector
α = (αi)di=1, and since there are q − 1 choices for each αi ∈ Fq\{0}, then there are at
least
cρd−1q(d2)(q − 1)d ≥ cρd−1q(
d+12 )
many distinct (d+ 1)-point configurations determined by E.
31
4.3 Proof of the second result (Theorem 4.2)
The following argument is adapted from an analogous argument, found in Section
6 of [8]. In what follows, we will prove the result for d = 2, then describe how to
generalize to higher dimensions, using a result from [9].
Proof. Suppose that a given set E ⊂ F2q has size |E| = n and has a corresponding
distance set ∆(E) = {‖x− y‖ ∈ Fq : x, y ∈ E} of size |∆(E)| = m. For any distance,
t ∈ ∆(E), find a pair of elements, a, b ∈ E, such that ‖a−b‖ = t. Now consider any of
the other (n−2) points, c ∈ E. Suppose that ‖a−c‖ = t1, and ‖b−c‖ = t2. Recalling
that circles in finite fields intersect at most twice, we have four possible intersections
of a circle of radius t1 and a circle of radius t2 centered at the points a and b. One of
these points is, by definition, c. Then there are at most three other points that can
make triangles with a and b that are congruent to the triangle abc. So the distance
represented by a and b determines (n − 2)/4 ≈ n distinct triangles. Starting with
t′ 6= t would give a different set of triangles. The overlap would be negligible, as
there are only three possible choices of the first distance that correspond to the same
triangle. This would imply that there are ≈ n different triangles for each of the ≈ m
distances, or ≈ nm = |E||∆(E)| distinct triangles.
To extend this argument to d-dimensions, we need to consider k different (k− 1)-
dimensional spheres. They can intersect in no more than 2k! places, unless more than
two of the centers lie on a line, more than three of their centers lie in a plane, etc.
Furthermore, consider the following theorem from [9], which will be used directly in
the proof of Theorem 4.2:
32
Theorem 4.7. Let E ⊂ Fdq, where |E| & qd+k2 for k ≤ d. Then |Tk(E)| & q(
k+12 ).
If we take k = d− 1, and apply Theorem 4.7 in the case of d-point configurations,
we find that E determines a positive proportion of d-point configurations so long as
|E| & qd−12 . Note that this is equivalent to requiring ρ & q−
12 , which is always in the
effective range of Theorem 4.2.
Now, consider any non-degenerate d-point configuration, T . Adding any other
point c ∈ E to our original configuration T defines a (d + 1)-point configuration T ′.
We know that d spheres centered at the points in T can only intersect in a constant
number of points. There are also a constant number of arrangements of the d distances
to c. As before, there are only a constant number of points that create (d+ 1)-point
configurations congruent to T ′. So, for each of the q(d2) d-point configurations, we can
find ≈ |E| ≥ ρqd distinct (d+ 1)-point configurations. So the total number of (d+ 1)-
point configurations that we are guaranteed is the number of d-point configurations
times the number of distinct (d+1)-point configurations that each of these are a part
of. In other words, we get
& q(d2)ρqd = ρq(
d+12 )
distinct (d+ 1)-point configurations.
33
Chapter 5
Dot products
5.1 Overview
In this chapter we discuss bounds on the number of distinct dot products determined
by large finite point sets in Rd. The first section is a simple pigeonholing argument
which gives a result in the plane. The second section gives an improvement to this
result using the Szemeredi-Trotter point-line incidence theorem. The third section
discusses a few special cases, which are related to the sets which may not be s-
adaptable for desired ranges of s, as briefly discussed in Subsection 3.6.2. The final
section uses the ideas of the first section to bootstrap the result from the second section
into higher dimensions. These should be compared with results of Steinerberger, in
[48]. Although, recently, Iosevich, Roche-Newton, and Rudnev, in [29], showed that
there must be more than n1−ε distinct dot products determined by any set of n points
in the plane, for any ε > 0.
5.2 The n12 argument
Given any x, y ∈ R2, we write their dot product as x·y. If x = (x1, x2) and y = (y1, y2),
x · y = x1y1 + x2y2.
34
There are other useful ways to think of the dot product, but this one will suffice for
the arguments to follow. Now, if we are given a set, P , of n points in the plane, define
Π(P ) to be the set of all distinct dot products determined by these points.
Theorem 5.1. Let P be a set of n points in the plane. Then
#Π(P ) & n12 .
Proof. First, pick a point out of P and call it x. Now, we know that the points whose
dot products with x are equal all lie on parallel lines. Suppose it takes t parallel
lines to cover our point set P . Now, if t &√n, we will have at least
√n distinct dot
products with x. What if t is significantly less than√n? By the pigeonhole principle,
we know that one of the lines, l, will have at least nt
points on it. Since we decided
that t <√n, we can be assured that n
t>√n. So we now have a line with
√n points
on it. Pick some point on l, that does not lie on the line through both x and the
origin, and call it y. Now, notice that covering the other points on l with another
set of parallel lines, each perpendicular to the line between y and the origin, gives us
√n populated lines. Recall that each of these lines represents a different dot product
with y. So either x or y will determine at least√n distinct dot products.
35
Figure 1: Drawing parallel lines that are perpendicular to the radial line through
the upper two points.
5.3 The n23 argument
First, we discuss the construction of a topological multigraph based on a point set,
and include an illustrative figure. If we are given a set, P , of n points, the first thing
we will do is construct a graph, G, similar to the one in the proof of Theorem 3.9.
So, define the vertex set to be the point set P . Draw the parallel lines which cover
the point set, which are perpendicular to the radial lines of points in our set. So after
drawing these parallel lines, perpendicular to the radial line of each point, for each
point, we will draw edges between consecutive points along these lines.
Now, suppose that there are several points r1, r2, ... along a particular radial line;
these points will all define the same set of parallel lines. First, we will draw edges
connecting consecutive points on these parallel lines because they each have the same
dot product with r1. Now, when we try to draw edges between consecutive points
along these parallel lines because they each have the same dot product with r2, we
36
have to connect the same pairs of points as before. As we process each of the points
on the radial line, we will have to draw an edge between the same pairs of consecutive
points as before. So we will have multiple edges between pairs of points along those
lines.
Figure 2: Consider the leftmost radial line. We draw parallel lines, each of which is
perpendicular to the leftmost radial line, which cover the points. The curved arcs
represent edges drawn between consecutive points on these parallel lines.
Now we prove the following theorem, which is the main result of this chapter.
Theorem 5.2. Let P be a set of n points in the plane. Then
#Π(P ) & n23 .
Proof. Let t be defined as the largest number of distinct dot products determined by
any point in P . We will assume that t . n23 , as otherwise, we are done. Now construct
G by letting the points in P act as vertices. For each point, x, in P , draw an edge
between pairs of consecutive points along the parallel lines which are perpendicular
to the radial line through x. Since we cover our point set with about n edges for each
37
point in P , there must be about n2 edges in G. Now, we can assume e ≈ n2, as if it
took fewer edges to connect the vertices, that would mean that there were many lines
which only had one point, and this would contradict our assumption that t . n23 .
If we consider a fixed radial line, l, the vertices of consecutive points along all
of the parallel lines perpendicular to l will be connected by as many edges as there
are points on l. We recall from the proof of Theorem 5.1 that there can be no more
than t points along any line much less on a radial line. So no pair of vertices can be
connected by more than t edges. Thus, in our graph, the maximum edge multiplicity
will be t.
We can apply the modified crossing number theorem, Theorem ??, to get:
n6
tn2.
e3
mv2. cr(G).
Now we just need an upper bound for the crossing number. Note that a crossing
between edges can only occur if a line perpendicular to one point’s radial line crosses a
line perpendicular to another point’s radial line. Since each point has fewer than t such
associated parallel lines, each pair of points can contribute at most t2 crossings. There
are about n2 different pairs of points so the total number of crossings is definitely less
than n2t2. So we can certainly bound the crossing number above by n2t2. Putting
the upper and lower bounds for the crossing number together:
n6
tn2. cr(G) . n2t2.
38
5.4 Special cases
In general, we had trouble reducing edge multiplicity. However, we can find some
special classes of sets where we can do better than in the general case. Below is an
odd looking theorem. It has a strange and seemingly artificial condition about the
number of points along a line through the origin. Soon enough though, we will see
how we can use a theorem like this to prove some interesting corollaries.
Theorem 5.3. Let #P = n. Also, let P and have no more than nx points on any
line through the origin. Then #Π(P ) & n1−x2 .
Proof. Recall that in the graph-theoretic proof of the dot product set result, we get
n6
mn2.
e3
mv2. cr(G) . n2t2.
Now, since no line through the origin has more than nx points on it, no edge mul-
tiplicity is higher than nx. So we can run the same argument with m = nx, and
get
t & n1−x2 ,
as claimed.
Suppose that you have two sets of real numbers, A and B. In the case that our
point set P can be expressed as a Cartesian product, A×B, we can gain over Theorem
5.2.
Corollary 5.4. Let P = A×B, where A,B ⊂ R, and #P = n. Let min(#A,#B) =
nx. Then #Π(P ) & n1−x2 .
39
Cartesian product sets come up quite often in practice, but this is not the only
kind of set that will obey the line condition in Theorem 5.3. There are plenty of
times where we want to deal with sets that are sufficiently spread out in some sense.
We introduce here a formal way to define a point set that is sufficiently spread out.
The following definition should be compared to the definition of s-adaptability, with
s = 2, in Chapter 3.
Definition 5.5. We call a set of size n, well-distributed well-distributed if it has
exactly one point inside each square of an n12 by n
12 lattice, where each square has
length C. The constant C can be any specified positive constant. For example, if
n = 100 and C = 1, then we could break up a square of area 100 into 100 unit
squares, each of which contains exactly one point.
Figure 3: Example of a well-distributed set with n = 9.
Corollary 5.6. Let P be well-distributed, and #P = n. Then
#Π(P ) & n34 .
40
Proof. Any line that passes through the set P may pass through no more than 2n12
squares. Thus, no line through the origin can pass through more than cn12 points. So
by Theorem 5.3,
#Π(P ) & n1− 12· 12 & n
34 .
The next definition is an example of a family of sets which can have very narrow
ranges of s for which they are s-adaptable.
Definition 5.7. We call a set of size n, 2-iterated well-distributed if it is comprised
of n12 translated well-distributed subsets, where each subset has constant C, and the
subsets are each contained in one square of an n14 by n
14 lattice of squares which have
side length max(C2, C−2). Similarly, a set is r-iterated well-distributed r-iterated well-
distributed if it is comprised of n1r translated (r−1)-iterated well-distributed subsets,
where each subset has constant max(Cr, C−r), and the subsets are each contained in
one square of an n12r by n
12r lattice of squares with side length max(Cr, C−r).
Figure 4: Example of a 2-iterated well-distributed set.
41
Note that, by the above definitions, well-distributed is the same as 1-iterated well-
distributed.
Corollary 5.8. Let #P = n, and let P be r-iterated well-distributed, where r ≤
log(n). Then #Π(P ) & n34 .
Proof. As in the proof of Corollary 5.6, the maximum number of large squares any
line can pass through is at most cn12r . Then, in each square, the maximum number
of subsquares any line can pass through is at most cn12r . This continues for r stages
of iterations, so the total number of points any line can pass through is certainly at
most cn12 .
5.5 Higher dimensions
Here, we extend Theorem 5.2 to higher dimensions by way of pigeonholing results from
lower hyperplanes embedded in space. In higher dimensions, if we fix a particular
dot product, s, and a point x, the points y such that x · y = s all lie on a hyperplane
orthogonal to the direction of x. Therefore, we can get higher dimensional results by
repeating an argument similar to the one in Section 5.2. In order to ease the following
exposition, we introduce a useful notion.
Definition 5.9. Let E ⊂ Rd. We define the pinned dot product set of E with respect
to a point x to be the set
Πx(E) = {x · y : y ∈ E} .
With this definition in hand, we prove a result in three dimensions.
Theorem 5.10. Given a set, P ⊂ R3 of n points, Π(E) & n25 .
42
Proof. Define t to be the largest number of distinct dot products determined by any
fixed point and the rest of P . Select some point x ∈ P such that #Πx(P ) = t. Recall
that each point which determines a particular dot product, s ∈ Πx(E), is confined to
a particular hyperplane, which we call Hx,s. There must be at least one hyperplane
with a maximal number points on it. Suppose it has m points, and is determined by
the dot product s0, and call it Hx,s0 . By the pigeonhole principle, m & nt.
For any point y on this hyperplane, whose direction vector is different from that
of x. Notice that #Πy(E) is again the number of hyperplanes, perpendicular to the
direction of y, which are populated by points in P . Now, these hyperplanes intersect
each of the hyperplanes Hx,s in lines confined to each Hx,s. The number of times a
particular dot product, r can occur between points in Hx,s0 is exactly the number of
incidences of points on Hx,s0 and a particular family of lines on it. These lines are
the intersections of Hx,s0 and Hy,r, where y 6= x.
Note that each one of these lines is unique to the point y 6= x. So the number of
times a given dot product can be determined by pairs of points in Hx,s0 is bounded
above by the number of incidences of m points and m lines in Hx,s0 . By Theorem
2.1, the number of incidences, I, is bounded above by
I . m23 .
Since there are about m2 possible pairs of points on Hx,s0 , the number of distinct dot
products determined by pairs of points on Hx,s0 is bounded below by
m2
m43
= m23 ≥
(nt
) 23.
So, we have that
Π(E) & t+(nt
) 23,
43
which guarantees that
Π(E) & n25 .
44
Chapter 6
Orthogonal vectors
6.1 Overview
In this chapter, we ask whether a sufficiently large subset of Fdq , contains a k-tuple
of mutually orthogonal vectors. Pairs of orthogonal vectors are studied in [1]. This
type of problem does not have a direct analog in Euclidean or integer geometries
because placing the set strictly inside {x ∈ Rd : xj > 0} immediately guarantees
that no orthogonal vectors are present. However, the arithmetic of finite fields allows
for a more complicated orthogonal structure. The main result of this chapter is the
following.
Theorem 6.1. Let E ⊂ Fdq, such that
|E| ≥ Cqdk−1k
+ k−12
+ 1k
with a sufficiently large constant C > 0, where
0 < (k2) < d.
Let λk be the number of k-tuples of k mutually orthogonal vectors in E. Then
λk = (1 + o(1))|E|kq−(k2).
45
Le Anh Vinh, in [52], showed a way to gain in the case k > 2 by employing
graph theoretic techniques that can be found in [1] and [35]. The threshold obtained
therein is |E| & qd2
+k−1, which admits a wider effective range for k in dimensions
greater than 2. The results coincide at k = 2, and are shown to be sharp in several
counterexamples below.
6.2 Proof of Theorem 6.1
Observe that
rk−1
(x1, ..., xk−1
)= q−(k−1)
∑si∈F∗q
i=1,2,...,k−1
∑xk∈Fdq
E(xk)k−1∏i=1
χ(−sixi · xk).
Lemma 6.2. ‖rk−1‖L2 . |E| 12 q(d−1)(k−1)+1
2 .
Assuming Lemma 6.2 for now, we prove the main result, Theorem 6.1.
Proof. Define Dk :={
(x1, ..., xk) ∈ Ek : xi · xj = 0,∀1 ≤ i < j ≤ k}
, where Ek means
E × E × ...× E︸ ︷︷ ︸k times
. Also, let Dk(x1, ..., xk) and E(x) be the indicator functions for the
set Dk and E, respectively. Clearly |Dk| = λk. Our goal is to get an expression for λk
in terms of λk−1. In order for that to do us any good, we will need an expression for
λ2. We will show the direct calculation of λ2, as well as the size condition on E for
two vectors. This will help to illustrate the ideas employed in the same calculations
46
for general k.
λ2 =∑
x1,x2∈Fdq :x1·x2=0
E(x1)E(x2)
= q−1∑
x1,x2∈Fdq
E(x1)E(x2)∑s∈Fq
χ(−sx1 · x2)
= q−1∑s∈Fq
∑x1,x2∈Fdq
E(x1)E(x2)χ(−sx1 · x2)
= I2 + II2,
where I2 is the sum over s = 0, and II2 is the same sum, but over s 6= 0. We will
show that I2 dominates the other term when |E| satisfies the size condition, and is
therefore the number of sets of 2 mutually orthogonal vectors present in E, modulo
a constant.
I2 =∑
x1,x2∈Fdq
E(x1)E(x2)q−1
= q−1∑x1∈Fdq
E(x1)∑x2∈Fdq
E(x2)
= |E|2q−1
If I2 indeed dominates the other two terms, we’ll have
λ2 = |E|2q−1.
So now we will have to compute II2. First we will separate the factors into the
indicator function of E and the discrepancy function. Then we will use Cauchy-
Schwarz so we can deal with the L2 norm of the discrepancy.
47
II2 = q−1∑s∈F∗q
∑x1,x2∈Fdq
E(x1)E(x2)χ(−sx1 · x2)
=∑
x1,x2∈Fdq
E(x1)q−1∑s∈F∗q
∑x2∈Fdq
E(x2)χ(−sx1 · x2)
≤ |E|12 (
∑x1,...,xk−1
r21)
12 ≈ |E|
12‖r1‖L2 .
Applying Lemma 6.2 gives us
‖r1‖L2 . |E|12 q
d2 .
So we can estimate II2 from above by |E|q d2 . Now we compare the sizes of I2 and
II2. Recall that we want our “main term”, I2, to dominate, so we get the expected
number of orthogonal pairs of vectors.
I2 > II2
|E|2q−1 > |E|q(d−1)+1
2
|E| > qd2
+1 = qd2−12
+ 2−12
+ 12 ,
as claimed. The same ideas work for higher k. In the general case, one must
operate with Dk−1 instead of the indicator function of E, and there is a product of
several additive characters present here, as opposed to only one. These and other
details are handled below.
48
λk =∑
xj∈Fdq :xj ·xk=0j=1,2,...,k−1
Dk−1(x1, ..., xk−1)E(xk)
= q−(k−1)∑xj∈Fdq
j=1,2,...,k
Dk−1(x1, ..., xk−1)E(xk)∑si∈Fq
i=1,2,...,k−1
k−1∏i=1
χ(−sixi · xk)
= q−(k−1)∑si∈Fq
i=1,2,...,k−1
∑xj∈Fdq
j=1,2,...,k
Dk−1(x1, ..., xk−1)E(xk)k−1∏i=1
χ(−sixi · xk)
= I + II + III,
where we separate the sum into three parts depending on the si’s. I is the sum
when all of the si’s are zero. II is the sum when none of the si’s are equal to zero.
III is the sum when some of the si’s are equal to zero, and some are not. We treat
these three cases separately. As before, we will show that I dominates the other terms
when |E| satisfies the size condition, and is a constant times the number of k-tuples
mutually orthogonal vectors contained in E.
I =∑xj∈Fdq
j=1,2,...,k
Dk−1(x1, ..., xk−1)E(xk)q−(k−1)
=∑xj∈Fdq
j=1,2,...,k−1
Dk−1(x1, ..., xk−1)|E|q−(k−1)
= |E|q−(k−1)∑xj∈Fdq
j=1,2,...,k−1
Dk−1(x1, ..., xk−1)
= |E|q−(k−1)λk−1
49
If I indeed dominates the other two terms, we’ll have
λkλk−1
= |E|q−(k−1).
To get an expression for λk, we recall the computation for k = 2 first: λ2 = |E|2q−1.
Then notice the following collapsing product.
λk =λkλk−1
· λk−1
λk−2
· · · · · λ3
λ2
· λ2.
Substituting each in ratio, as computed above, yields
λk =|E|k
q(k2).
Now we need to compute II, the biggest error term. Now we recall the definition
of the discrepancy function.
rk−1
(x1, ..., xk−1
)= q−(k−1)
∑si∈F∗q
i=1,2,...,k−1
∑xk∈Fdq
E(xk)k−1∏i=1
χ(−sixi · xk)
First, we dominate the sum of the xj’s in E by the sum of xj’s in all of Fdq . Then
we apply Cauchy-Schwarz to the sum over the xj’s.
II = q−(k−1)∑si∈F∗q
i=1,2,...,k−1
∑xj∈Fdq
j=1,2,...,k
Dk−1(x1, ..., xk−1)E(xk)k−1∏i=1
χ(−sixi · xk)
≤∑xj∈Fdq
j=1,...,k−1
Dk−1(x1, ..., xk−1)q−(k−1)∑si∈F∗q
i=1,...,k−1
∑xk∈Fdq
E(xk)k−1∏i=1
χ(−sixi · xk)
≤ λ12k−1(
∑x1,...,xk−1
r2k−1)
12 ≈ |E|
k−12 q
−(k−12 )
2 ‖rk−1‖L2 .
50
So we use Lemma 6.2 to get a handle on ‖rk−1‖L2 . Now we are guaranteed that
II . |E|k−12 q
−(k−12 )
2 |E|12 q
(d−1)(k−1)+12 .
= |E|k2 q
(d−1)(k−1)+1−(k−12 )
2
To deal with III, break it up into sums that have the same number of non-zero
sj’s.
III =∑
one sj=0
+∑
two sj ’s=0
+...
= d∑s1=0
+d(d− 1)∑
s1=s2=0
+...
Now each of these sums will look like II, but with (k − 2) instead of (k − 1) for
the first sum, and (k−3) instead of (k−1) in the second sum, and so on. This allows
us to bound each sum in III as follows:
III . d|E|k−22 q
(k−22 )2 ‖rk−2‖L2 + d(d− 1)|E|
k−32 q
(k−32 )2 ‖rk−3‖L2 + ...
So III is dominated by II as long as q > d, which is guaranteed, as q grows
arbirtarily large.
Now we only need to find appropriate conditions on E to ensure that I > II.
I > II
|E|kq−(k2) > |E|k2 q
(d−1)(k−1)+1−(k−12 )
2
|E|k2 > q
2(d−1)(k−1)+2−(k−1)(k−2)+2k(k−1)4
|E| > q(k−1k )d+ k−1
2+ 1k .
51
Now we prove the Lemma 6.2.
Proof. Recall the definition of rk−1
(x1, ..., xk−1
)and use orthogonality in x1, ..., xk−1.
‖rk−1‖2L2 = q−2(k−1)
∑x1,...,xk−1
∑s1,s′1,...,sk−1,s
′k−1
∑xk,yk∈E
k−1∏j=1
χ((sjxk − s′jyk) · xj)
= qd(k−1)q−2(k−1)
∑sj=s′j
∑xk,yk:
sjxk=s′jy
k
E(xk)E(yk) +∑sj 6=s′j
∑xk,yk:
sjxk=s′jy
k
E(xk)E(yk)
= q(d−2)(k−1) (A+B)
Let us approach A first. Since s1 = s′1, and sjxk = s′jy
k for all j, we know that it
holds for j = 1, and therefore xk = yk. This tells us that sj = s′j for all j. So
A =∑
s1,...,sk−1
∑xk
E(xk)E(xk) = q(k−1)∑xk
E(xk)E(xk) = |E|q(k−1)
Now we tackle the quantity B. Here we introduce a new variable, α = s1s′1
. We know
that s′j 6= 0, as they are elements of F∗q. Also notice that the condition sjxk = s′jy
k
implies α =sjs′j
for all j. So we did have to sum over 2(k − 1) different variables, but
now we know that these are completely determined by only (k−1) of the originals. So
we will have (k − 1) free variables. In light of this, with a simple change of variables
we get
B = q(k−1)∑
yk=αxk
E(xk)E(αxk)
≤ q(k−1)∑xk∈Fdq
|E ∩ lxk |
≤ |E|qk
52
Where, lxk :={txk ∈ Fdq : t ∈ Fq
}, which can only intersect E at most q times.
With the estimates for A and B in tow,
‖rk−1‖2L2 = q(d−2)(k−1) (A+B)
. q(d−2)(k−1)(|E|q(k−1) + |E|qk
)≈ |E|q(d−1)(k−1)+1.
6.3 Sharpness examples
The following lemmata are included to show how close Theorem 6.1 is to being sharp.
There are several possible notions of sharpness for this result. It is clearly interesting
to consider how big a set can be without containing any orthogonal k-tuples. The
first lemma is merely an intuitive construction used in the next lemma, both of which
concern large sets with no orthogonal k-tuples. The last lemma is included to show
how close our size condition is if we relax the allowable number of orthogonal k-tuples.
Lemma 6.3. There exists a set E ⊂ F2q such that |E| ≈ q2, but no pair of its vectors
are orthogonal.
Proof. This is done by taking the union of about q2
lines through the origin, such
that no two lines are perpendicular, and removing the union of their q2
orthogonal
complements, which are lines perpendicular to lines in the first union. Then our set
E has about q2
2points, but no pair has a zero dot product.
53
The next result is the main counterexample, which shows that it is possible to
construct large subsets of Fdq with no pairs of orthogonal vectors.
Lemma 6.4. There exists a set E ⊂ Fdq such that |E| ≥ cqd2
+1, for some c > 0, but
no pair of its vectors are orthogonal.
Proof. The basic idea is to construct two sets, E1 ⊂ F2q, and E2 ⊂ Fd−2
q , such that
|E1| ≈ q32 and |E2| ≈ q
d−12 . If you pick q and build these sets carefully, you can
guarantee that the sum set of their respective dot product sets does not contain 0.
The following algorithm was inspired by [24].
Here we will indicate how to construct E1. The construction of E2 is similar. First,
let q = p2, where p is a power of a large prime. We also pick these such that p+ 1 is
of the form 4n, where n is odd. This way we can be guaranteed a large, well-behaved
multiplicative group of order q − 1 = (p− 1)(p+ 1), as well as a subfield of order p.
Let i denote the square root of −1, which is in F∗q, since q is congruent to 1 mod
4. Now let B be a cyclic subgroup of F∗q of order p+14
(p− 1) = n(p− 1). Since n was
odd, and p was congruent to 3 mod 4, we know that 4 does not divide the order of B.
This means that B has no element of order 4, so it is clear that i /∈ B. Let β denote
the generator of B, as it is a subgroup of a cyclic group, and therefore cyclic. Since
p− 1 is even, we know that we can find another cyclic subgroup, A, generated by β2.
Let Cp be the elements of F∗p that lie on the unit circle, that is,
Cp :={x ∈ F2
p : x21 + x2
2 = 1}.
From a lemma in [24], (or basic number theory) we know that |Cp| = p− 1, since
−1 is not a square in a field of order congruent to 3 mod 4. We can be sure that for
54
all u, v ∈ Cp, u · v ∈ Fp. Now let
E ′1 := {τu : τ ∈ A, u ∈ Cp} .
So, for all x, y ∈ E ′1, we can be sure that x · y ∈ A ∪ {0}. To see this, let x = σu,
and y = τv, where σ, τ ∈ A and u, v ∈ Cp. Then x · y = στ(u · v) ∈ A ∪ {0}, as any
non-zero u · v ∈ F∗p ⊂ A. Now, the cardinality of E ′1 is
|E ′1| = |Cp||A| = (p− 1)
(p+ 1
4
p− 1
2
)≈ q
32 .
Now pick q2
mutually non-orthogonal lines in E ′1. Call this collection of lines L.
Let L⊥ indicate the set of lines perpendicular to the lines in L. Now we need to prune
E ′1 so that it has no orthogonal vectors. One of the sets E ′1 ∩ L or E ′1 ∩ L⊥ has more
points. Call the set with more points E1. This means that no zero dot products can
show up in E1, in a similar manner to the construction in the proof of Lemma 6.3.
Now we have |E1| ≈ q32 , and for any x, y ∈ E1, we are guaranteed that x · y ∈ A,
which does not contain 0.
Construct E2 ⊂ Fd−2q in a similar manner, using spheres instead of circles. How-
ever, in the construction of E2, we do not need to prune anything. Now we have
|E2| ≈ qd−12 and all of its dot products lie in A ∪ {0}. Set E = E1 × E2. Since
E1 has its dot product set contained in A, and E2 has its dot product set contained
in A ∪ {0}, we know that any dot product of two elements in E is in the sum set
A+ (A ∪ {0}).
Now we will show that 0 is not in the dot product set. If two elements did have
a zero dot product, that would mean that we had s, t ∈ A, where s comes from the
first two dimensions, or E1, and t comes from the other d− 2 dimensions, or E2, and
55
we also have s = −t. (Note, even though t could conceivably be zero, s can not, so
we would not have s = −t if t were zero. Therefore t is necessarily an element of A.)
Recall that s and t are squares of elements in B. Call them σ2 and τ 2, respectively,
for some σ, τ ∈ B. Since B has multiplicative inverses, let α = στ∈ B. So we would
need the following:
σ2 = −τ 2 ⇒ −1 =σ2
τ 2= α2.
But we constructed B so that it does not contain the square root of −1. Therefore
there can be no two elements of E which have a zero dot product.
The authors believe that the preceeding example can be generalized to obtain
results about how large a set can be without containing orthogonal k-tuples for k > 2.
The next example is trivial, but is included to indicate one way in which we get some
orthogonal vectors, but not as many as would be expected by initial considerations.
Lemma 6.5. There exists a set E ⊂ Fdq such that |E| ≈ qk−1k
(d−1)+ k−12
+ 1k , but only
q(k−1)(d−1)+1 k-tuples of its vectors are orthogonal.
Proof. Consider a set E1 ⊂ Fd−1q that satisfies the size condition for d − 1 dimen-
sions. That is, |E1| ≈ qk−1k
(d−1)+ k−12
+ 1k . By Theorem 6.1, we know that E1 has about
|E|kq−(k2) ≈ q(k−1)(d−1)+1 different k-tuples of mutually orthogonal vectors. If we con-
sider E = E1 × {0} ⊂ Fdq , we have |E| = |E1| ≈ qk−1k
(d−1)+ k−12
+ 1k , but we have only
about q(k−1)(d−1)+1 different k-tuples of mutually orthogonal vectors.
The set constructed in Lemma 6.5 has fewer than the “statistically expected”
number of mutually orthogonal vectors with respect to the dimension d. Considering
56
only the size of the set, this example shows that if we lose a factor of about qk−1k points
in our set, we lose a factor of about qk−1 k-tuples of mutually orthogonal vectors. Of
course, this is quite artificial, in the sense that E “really” lives in a smaller dimension,
but it does indicate that the result is relatively sharp. This also points to a rather
intuitive measure of dimension of a set, if one has a handle on the number of mutually
orthogonal k-tuples.
57
Chapter 7
Sharpness examples related to theunit distance problem
7.1 Overview
We recall that the classical Falconer distance conjecture ([17]) says that if the Haus-
dorff dimension of a compact set E ⊂ Rd, d ≥ 2, is greater than d2, then the Lebesgue
measure of
∆(E) = {|x− y| : x, y ∈ E}
is positive. Bourgain ([6]), Erdogan ([14]), Mattila ([40], [41]), Wolff ([57]) and others
have improved the exponent, with the best current result due to Wolff in two dimen-
sions, in [57], and Erdogan in three and more dimensions, in [14]. They have shown
that that L1(∆(E)) > 0 provided that dimH(E) > d2
+ 13. In a related line of study,
[42], the authors prove that if dimH(E) > d+12
, then ∆(E) contains an interval.
As mentioned in Chapter 2, Falconer’s d+12
exponent follows from (2.2), which we
recall below. Suppose that µ is Borel measure on E such that
Is(µ) =
∫ ∫|x− y|−sdµ(x)dµ(y) <∞
for every s > d+12
. Then we get (2.2), that
µ× µ{(x, y) : 1 ≤ |x− y| ≤ 1 + ε} . ε.
58
We draw a parallel between this estimate and the unit distance problem from the
discrete setting. Basically, what we are asking for here is that no distance occur
too frequently. Clearly, if this estimate held for any s > d2, the Falconer distance
problem would be solved, but the resolution of the Falconer distance problem will not
immediately imply anything about the estimate (2.2).
In the regime where s > d+12
, this estimate follows by Plancherel and the fact that
|σ(ξ)| . |ξ|−d−12 , (7.1)
where σ denotes the Lebesgue measure on the unit sphere.
If we replace the Euclidean distance, | · |, with a new metric, || · ||B, where B
is a symmetric convex body with a smooth boundary and everywhere non-vanishing
Gaussian curvature, instead of the unit sphere, then (2.2) still holds. The main reason
for this is that the stationary phase estimate, (7.1), still holds if σ is replaced by σB,
the Lebesgue measure on the surface ∂B. In other words, if B is a symmetric convex
body with everywhere non-vanishing Gaussian curvature, the estimate
µ× µ{(x, y) : 1 ≤ ||x− y||B ≤ 1 + ε} . ε (7.2)
holds whenever Is(µ) <∞ with s > d+12
.
A consequence of this more general version of (2.2) is that L1(∆B(E)) > 0 when-
ever dimH(E) > d+12
, where
∆B(E) = {||x− y||B : x, y ∈ E}.
An example due to Mattila (see [39]) shows in two dimensions that for no s < 32
does
Is(µ) =
∫ ∫|x− y|−sdµ(x)dµ(y) <∞
59
imply (7.2). Mattila’s construction can be generalized to three dimensions. However,
in dimensions four and higher, his method does not seem to apply. It is important to
note that in any dimensions, an example due to Falconer ([17]) shows that for no s < d2
does Is(µ) <∞ imply that the estimate (7.2) hold. We record these calculations for
the reader’s convenience in the Section 7.3 below.
In this chapter, we construct a measure in all dimensions which shows that for no
s < d+12
does Is(µ) <∞ imply that (7.2) hold. More precisely, we have the following
result.
Theorem 7.1. There exists a symmetric convex body B with a smooth boundary and
non-vanishing Gaussian curvature, such that for any s < d+12
, there exists a Borel
measure µs, such that Is(µ) ≈ 1 and
lim supε→0
ε−1µs × µs{(x, y) : 1 ≤ ||x− y||B ≤ 1 + ε} =∞. (7.3)
Remark 2. The proof will show that ε−1 in (7.3) may be replaced by ε−2sd+1−γ for any
γ > 0. We also note that we only need to establish (7.3) with s ≥ d2
since if s < d2,
the example due to Falconer ([17]), mentioned above, does the job.
Another way of stating the conclusion of Theorem 7.1 is that for no s < d+12
does
Is(µ) < ∞ imply that the distance measure is in L∞(R). The distance measure ν is
defined by the relation
∫g(t)dν(t) =
∫ ∫g(||x− y||B)dµ(x)dµ(y).
Theorem 7.1 is proved in Section 7.2 below. The idea is to make a construction
for a specific convex body obtained by gluing the upper and lower hemispheres of
the paraboloid. In the Subsection 7.2.1 we describe the combinatorial construction
60
used in the proof of Theorem 7.1. We also describe another construction, due to Lenz
and point out that it cannot be used in our setting. The proof of this assertion is
postponed to Section 7.4. In Subsection 7.2.2 we use the combinatorial construction
from Section 7.2.1 to complete the proof of Theorem 7.1. In Section 7.3 we describe
an example due to Mattila and generalize it three dimensions. In Section 7.4 we
explain the connection between the estimates (2.2) and (7.2) and how this connection
rules out the Lenz construction. Finally, in Section 7.5, we explore related results in
vector spaces over Zq, the integers mod q.
7.2 Proof of the main result
7.2.1 Combinatorial underpinnings
The proof of Theorem 7.1 uses a generalization of the two-dimensional construction
due to Pavel Valtr (see [8], [56]). A similar construction can also be found in [34] in
a slightly different context. Let
Pn =
{(i1n,i2n, . . . ,
id−1
n,idn2
): 0 ≤ ij ≤ n− 1, for 1 ≤ j ≤ d− 1, and 1 ≤ id ≤ n2
}.
Notice that in each of the first d−1 coordinates, there are n evenly distributed points,
but in the last dimension, there are n2 evenly distributed points. Now, let
H = {(t, t, . . . , t, t2) ∈ Rd : t ∈ R}
and define
LH = {H + p, p ∈ Pn}.
61
Figure 5: On the left, we see a picture of the set P5, on the right, we see it again
with a few parabolic arcs, which intersect a point in each column.
Let N = nd+1. By construction, #Pn = N , which is also the number of translates
of H in LH . Also by construction, each element of LH is incident to about nd−1 ≈
Nd−1d+1 elements of P . Thus the total number of incidences between P and LH is
≈ N1+ d−1d+1 = N
2dd+1 = N2− 2
d+1 .
Construction of the norm
With this construction in hand, it is easy enough to flip the paraboloid upside down
and glue it to another copy. Explicitly, let
BU =
{(x1, . . . , xd) ∈ Rd : xi ∈ [−1, 1], for 1 ≤ i ≤ d− 1, xd = 1−
(x2
1 + · · ·+ x2d−1
)},
and
BL =
{(x1, . . . , xd) ∈ Rd : xi ∈ [−1, 1], for 1 ≤ i ≤ d− 1, xd = −1 + x2
1 + · · ·+ x2d−1
}.
Now, let
B′ =(BU ∩
{(x1, . . . , xd) ∈ Rd : xd ≥ 0
})∪(BL ∩
{(x1, . . . , xd) ∈ Rd : xd ≤ 0
}).
62
Finally, define B to be the convex body B′, with the ridge at the transition between
BU and BL smoothed.
Let L denote N copies of ∂B, each translated by an element of Pn. Now we have
a symmetric convex body B ⊂ Rd with a smooth boundary and everywhere non-
vanishing curvature, a point set Pn of size N and a set L of translates of ∂B, of size
≈ N , such that the number of incidences between Pn and L is ≈ N2− 2d+1 .
The reader may be aware of the fact that in dimensions four and higher, a more
dramatic combinatorial example is available.
Lenz construction
(see e.g. [8]) More precisely, choose N/2 points evenly spaced on the circle
{(cos(θ), sin(θ), 0, 0) : θ ∈ [0, 2π)}
and N/2 points evenly spaced on the circle
{(0, 0, cos(φ), sin(φ)) : φ ∈ [0, 2π)}.
Let KN be the union of the two point sets. It is not hard to check that all the
distances between the points on one circle and the points on the other circle are equal
to√
2. It follows that the number of incidences between the points of KN and the
circles of radius√
2 centered at the points of KN is ≈ N2, which is about as bad as
it can be and much larger than the N2− 2d+1 obtained in the generalization of Valtr’s
example above. However, this construction will not help in the continuous setting
due to certain peculiarities of the Hausdorff dimension. This, in turn, leads to some
interesting combinatorial questions. We shall discuss this issue in the Section 7.4.
63
7.2.2 Measure construction using combinatorial information
Let d2≤ s < d+1
2. Recall that there is no point going below d
2because the lattice-based
construction in [17] shows that (2.2) cannot hold in that regime. Partition [0, 1]d into
lattice cubes of side-lengths ε where ε−s = N for some large integer N . Let n = N1d+1 .
Put Pn in the unit cube and select any lattice cube which contains a point of Pn. Let
Qn denote the set of centers of the selected lattice cubes.
Now, we define Lε to be the union of the ε-neighborhoods of the elements of L.
That is, for every translate of ∂B by an element, p ∈ Pn, let lp denote the locus of
points that are within ε of the translate of ∂B by p. Then
Lε =⋃p∈Pn
lp
.
Figure 6: On the left, we have P5, in the partitioned unit cube. On the right, we
filled in every cube which contained a point. Notice that there are gaps in the
columns corresponding to cubes which did not contain any points.
Lemma 7.2. Let µs denote the Lebesgue measure on the union of the selected cubes
above, normalized so that ∫dµs(x) = 1.
64
More precisely,
dµs(x) = εs−d∑p∈Qn
χRε(p)(x)dx,
where Rε(p) denotes the cube of side-length ε centered at p. Then
Is(µs) ≈ 1.
Proof. To prove the lemma, observe that
Is(µ) =
∫ ∫|x− y|−sdµs(x)dµs(y)
= ε2(s−d)∑p,q∈Pn
∫ ∫|x− y|−sχRε(p)(x)χRε(q)(y)dxdy
= ε2(s−d)∑p∈Pn
∫ ∫|x− y|−sχRε(p)(x)χRε(p)(y)dxdy
+ε2(s−d)∑
p 6=q∈Pn
∫ ∫|x− y|−sχRε(p)(x)χRε(q)(y)dxdy = I + II.
We have
I = ε2(s−d)∑p∈Pn
∫Rε(p)
∫Rε(p)
|x− y|−sdxdy.
Making the change of variables X = x− y, Y = y, we see that
I . ε2(s−d)∑p∈Pn
εd∫|X|≤
√dε
|X|−sdX
. ε2(s−d) · εd · εd−s∑p∈PN
1
. εs ·N . 1.
On the other hand,
II ≈∑
p 6=q∈Pn
|p− q|−sε2s
= N−2∑
p6=q∈Pn
|p− q|−s.
65
We have
p = (p′, pd) =
(i1n, . . . ,
id−1
n,idn2
)and
q = (q′, qd) =
(j1n, . . . ,
jd−1
n,jdn2
).
Let i′ = (i1, . . . , id−1) and j′ = (j1, . . . , jd−1).
Thus we must consider
N−2∑
i 6=j;|i′|,|j′|≤n;id,jd≤n2
∣∣∣∣∣∣∣∣i′ − j′n
∣∣∣∣+
∣∣∣∣id − jdn2
∣∣∣∣∣∣∣∣−s.Replacing the sum by the integral, we obtain
N−2
∫ ∫. . .
∫|i1|,|j1|,...,|id−1|,|jd−1|≤n
id,jd≤n2
∣∣∣∣∣∣∣∣i′ − j′n
∣∣∣∣+
∣∣∣∣id − jdn2
∣∣∣∣∣∣∣∣−sdi′dj′diddjd,which, by a change of variables, u′ = (i′/n), ud = (id/n
2), v′ = (j′/n), and vd =
(jd/n2), with similarly named coordinates, becomes
≈∫ ∫
. . .
∫u6=v
|u1|,|v1|,...,|ud−1|,|vd−1|≤1ud,vd≤1
|u− v|−sdu′dv′duddvd . 1.
This completes the proof of Lemma 7.2.
We are now ready to complete the argument in the case of the paraboloid. We
have
µs × µs{(x, y) : 1 ≤ ||x− y||B ≤ 1 + ε}
is≈ Cε2s times the number of incidences between the elements of Qn and Lε, where Qn
and Lε are constructed in the beginning of this section. Invoking our generalization
of Valtr’s construction from Section 7.2.1 above, we see that
µs × µs{(x, y) : 1 ≤ ||x− y||B ≤ 1 + ε} ≈ ε2s ·N2− 2d+1 ≈ N−
2d+1 .
66
This quantity is much greater than ε = N−1s when s < d+1
2. This completes the
proof of Theorem 7.1.
7.3 Mattila’s construction
In this section we describe Mattila’s construction from [39] and its generalization to
three dimensions.
First, we review the method of constructing a Cantor set of a given Hausdorff
dimension, (see [41]). If we want a Cantor set, Cα, of Hausdorff dimension 0 < α < 1,
we need to find the 0 < λ < 1/2 which satisfies α = log 2/ log(1/λ). Start with the
unit segment, then remove the interval (λ/2, 1−λ/2) from the middle. Next, remove
the middle λ proportion of each of the remaining subintervals, and so on. The classic
“middle-thirds” Cantor set would be generated with λ = 1/3.
To construct the two-dimensional example, M2(α), we let F = (Cα) ∪ (Cα − 1).
Then define M2(α) = F × [0, 1]. Define the measure µ to be (Hα|F ) × (L1|[0, 1]) ,
where Hα is the α-dimensional Hausdorff measure.
Pick a point x = (x1, x2) ∈ M2(α). Notice that if x1 ∈ F , either x1 + 1 or x1 − 1
is also in F . So there is an ε-annulus, with radius 1, centered at x, which contains a
rectangle of width ≈ ε and length ≈√ε. This rectangle intersectsM2(α) lengthwise.
The measure of this intersection is ε1/2+α. This follows easily from the fact that the
circle has non-vanishing curvature. It follows that
µ {y : 1 ≤ |x− y| ≤ 1 + ε} & εα+1/2
67
for every x. It follows that
µ× µ{(x, y) : 1 ≤ |x− y| ≤ 1 + ε}
=
∫µ {y : 1 ≤ |x− y| ≤ 1 + ε} dµ(x)
& εα+1/2.
We conclude that
µ× µ{(x, y) : 1 ≤ |x− y| ≤ 1 + ε} . ε
only if
εα+ 12 . 1,
which can only hold if
α ≥ 1
2.
Thus the estimate (7.2) does not in general hold for sets with Hausdorff dimension
less than 32. Letting α get arbitrarily small yields a family of counterexamples with
Hausdorff dimensions arbitrarily close to 1 , below which there are already counterex-
amples. See, for example, [17]. Note that we worked in [−1, 1] × [0, 1] instead of
[0, 1]× [0, 1], to allow the main point to shine.
To construct M3(δ), the three-dimensional example, we set
M3(δ) = (Cα ∪ Cα − 1)× (Cα ∪ Cα − 1)× Cβ,
where α = 1 − δ, and β = δ/2, and δ is determined later. We will set µ to be a
product of the appropriate Hausdorff measures restricted to this set, much like the
previous example. Notice that M3(δ) has a Hausdorff dimension of 2 − 32δ, and for
68
a given point x ∈ M3(δ), there is an ε12 by ε
12 by ε box inside the annulus whose
measure is
εα/2 · εα/2 · εβ = ε1−δ/2.
Once again, we have used the fact that the sphere has non-vanishing Gaussian
curvature, which implies, by elementary geometry, that the ε-annulus contains and
ε12 by ε
12 by ε box. It follows that
µ {y : 1 ≤ |x− y| ≤ 1 + ε} & ε1−δ/2
for every x, which means that
µ× µ{(x, y) : 1 ≤ |x− y| ≤ 1 + ε} & ε1−δ/2,
so (7.2) does not hold.
Thus we shown that for s < 2 = d+12
(when d = 3), Is(µ) <∞ does not imply that
(7.2) holds. Obvsere that both constructions in this sections work for any convex B
such that ∂B is smooth and has everywhere non-vanishing Gaussian curvature.
In Section ?? below we shall see that Mattila’s example can be used to construct
a discrete point set and a family of annuli such that the number of incidences is much
greater than it is in the corresponding problem where the point set is a lattice.
It is interesting to note that Mattila’s example can be adapted (see [10]) to study
the sharpness of the following generalization of the Falconer estimate:
µ× · · · × µ{(x1, . . . , xk+1) : tij ≤ ||xi − xj||B ≤ tij + ε} . ε(k+12 ).
This estimate is used to study k-simplices in the way that the Falconer’s esti-
mate is adapted to study distance, which may be viewed as 1-simplices, or two-point
69
configurations.
7.4 Inequality (7.2) and the discrete incidence the-
orem
The reason we are able to use a generalization of Valtr’s construction and are unable
to use the Lenz construction described in subsubsection 7.2.1 is due to Lemma 7.2.
Roughly speaking, the idea is the following. We want to take a discrete set of points
in Rd and turn it into a set which behaves as though it had Hausdorff dimension s > 0
by thickening each of the N points by N−1s . The resulting set has positive Lebesgue
measure, of course, but the measure tends to 0 as N approaches infinity and we may
view this set as s-dimensional if the energy integral
Is(µ) =
∫ ∫|x− y|−sdµ(x)dµ(y) ≈ 1,
where dµ is the Lebesgue measure on the resulting union of N−1s balls, normalized
so that ∫dµ(x) = 1.
Hausdorff dimension captures more the just the amount of mass in a set; it also
measures the distribution of mass within the set. See [26], [28], and [27] in the
context of homogeneous or Delone sets. For more generality, see [31]. Notice that the
s-adaptability condition holds for our adaptation of the Valtr construction, but not
the Lenz construction.
Proposition 7.3. The Lenz construction, KN , is not s-adaptable for s > 1.
Proof. Let K ′N denote the points on the circle in the first quadrant of the first two
70
dimensions.
N−2∑
p,q∈KN
|p− q|−s & N−2∑
p,q∈K′N
|p− q|−s
& N−2∑p∈K′N
∑q∈K′N|p−q|≤1/2
|p− q|−s
& N−2N
N/6∑j=1
(j
2N
)−s
& N s−1
N/6∑j=1
j−s
& N s−1,
where, in the second inequality, when we fix a point p, we measure the distance to
points q which are close enough, so that in the next inequality, these distances are at
least one-half of j/N , where the point q is j points away from p along the circle. The
quantity N s−1 is unbounded as N grows large for s > 1.
Notice that the proof did not rely on the ambient dimension at all. This raises
the possibility of studying the Erdos single distance conjecture in higher dimensions
under the assumption that the set is s-adaptable. This definition may be viewed as
a natural generalization of the notion of homogeneous sets, used, for example, by
Solymosi and Vu in [45].
One can adapt the techniques in [27], originally applied to homogeneous sets, to
prove the following result.
Theorem 7.4. Suppose that PN is s-adaptable with s ≥ d+12
. Then for any t > 0,
#{(x, y) ∈ PN : ||x− y||B = t} . N2− 1s . (7.1)
71
for any convex body B with a smooth boundary and everywhere non-vanishing Gaus-
sian curvature.
The question is whether one can push the range of exponents s for which (7.1)
holds below d+12
. Valtr’s example shows that this is not possible for general metrics,
even the ones generated by norms where the boundary of the underlying convex body
is smooth and has non-vanishing Gaussian curvature.
7.5 Valtr’s example applied to vector spaces over
Zq
Here we show that Valtr’s construction easily extends to vector spaces over Zq. We
will construct a subset of a vector space over Zq, and a set of “surfaces” associated
to each point in the set, such that there are more than the “statistically expected”
number of incidences.
A common analog of the Euclidean distance in Zdq , the d-dimensional vector space,
is |x− y| = (x1− y1)2 + ...+ (xd− yd)2. This is because the square root may not be an
integer. In [30], it is shown that if #E & q(d+1)/2, then #∆(E) & q. To prove this,
they showed that the number of incidences between points in the set and “spheres” of
fixed radii was (#E)2/q+D(q), where the discrepancy term, D(q), is bounded above
by #E · q(d−1)/2. Later, in [?], it was shown that there exist subsets of Fdq , with d odd,
where #∆(E) 6= Fq. In what follows, we state results for Zdq instead of Fdq , but these
are very closely related settings. The following theorem is an analog of Theorem 7.1
for this setting.
Theorem 7.5. There exists a family of sets E ⊂ Zdq, with #E � qd2 , and a family
of related quadratic surfaces for which there are more than the statistically expected
72
number of incidences.
Proof. To construct the set, E ⊂ Zdq , fix a small parameter, ε > 0. Let k Be cho-
sen later. Let δ, l, and n satisfy qδ = 2k, qε = 2l, and q = 2n. Define A to be
{1, 2, ..., q1/2−δ = 2n/2−k}. Now, let B be the set of squares of elements in A, or
{a2 : a ∈ A}. Notice that for any b ∈ B, b requires no more than (n − 2k) bits to
be written in binary. Now, let A = {1, 2, ..., q1−ε = 2n−l}. Finally, define E to be
A× A× ...× A×A, where there are (d− 1) copies of A.
The family of surfaces will be Γa1,...,ad−1,α = {(t1 + a1, . . . , td−1 + ad−1, t21 + · · · +
t2d−1 + α) : tj ∈ A}. For every Γ, each of the first (d− 1) coordinates will take values
in A+A = {a+a′ : a, a′ ∈ A}. By the definition of the set A, #((A+A)∩A) & #A.
So we can be assured that a positive proportion of the Γ have the property that each
of their first (d − 1) coordinates are in A. Now, notice that t21 + · · · + t2d−1 has no
more than (n − 2k + (d − 2)) bits, as it is a sum of (d − 1) elements with no more
than (n− 2k) bits each. We want to be certain that for a positive proportion of the
Γ, namely those whose α has fewer than (n − 2k + d) bits, a positive proportion of
the values of the dth coordinate will fall in A. We can guarantee this if l ≤ 2k − d.
This means that the for a positive proportion of the Γ, we will get about (#A)d−1
incidences. Since there is a different Γ for every point in E, we get that the number
of incidences is about #E · (#A)d−1 = (#A)2(d−1) ·#A.
We now look for a range on δ in which the number of incidences is much greater
than the expected number of incidences, or (#E)2/q. This holds precisely when
(#A)2(d−1) · #A � (#E)2/q, or whenever ε > c > 0. However, in order to assure
that our set is non-trivial, it must have size greater than qd/2, which is satisfied when
73
ε < 1/2− δ(d− 1), or equivalently, when k < (n/2 + (d− 2))/(d+ 1).
74
Chapter 8
Applied results
8.1 Overview
This chapter features two types of results which tackle the general problem of trans-
mitting information through noisy media. The first example comes from frame theory.
A frame is basically a spanning set of vectors for some space. Often these sets are
over-complete, that is, there are more vectors than necessary to span the given space.
This redundancy allows for excellent error mitigation with minimal computation as
we will see. The second example comes from the area of coding theory. Here, we give
explicit bounds for the existence of check matrices, which are a fundamental element
of a large class of codes.
8.2 Semi-circular frames
Suppose HK is a K-dimensional Hilbert space. Then any spanning set of vectors,
F = {fj ∈ HK}N=1 can be considered a frame. If all of the vectors in a frame have
the same norm, then we call it an equal-norm frame. We define the analysis operator
of a frame as V : HK → `(2) by V (X) = {〈x, fj〉}Nj=1. The adjoint of this operator is
called the synthesis operator. It satisfies V ∗({ai}Ni=1) =∑N
i=1 aifi. The frame operator
75
is defined as
V ∗V (X) =N∑i=1
〈x, fi〉fi,
and is a positive, self-adjoint operator on HK . If V ∗V = AI, for some constant A, we
call F an A-tight frame. A 1-tight frame is called a Parseval frame.
Often, frames have more vectors than necessary to span the associated Hilbert
space. This makes them ideal candidates for methods of transferring data through
noisy media. We define one type of frame, introduced by Bodmann and Paulsen in
[3] which has proved to be particularly useful.
Definition 8.1. A frame F in a Hilbert space HK is called a semi-circular frame if
its vectors are evenly spaced along semicircles in each plane determined by a pair of
orthogonal vectors from HK .
In a typical application, frame vectors are pre-determined and fixed, while the
coefficients of these vectors vary over time, and are used to transmit data. One
prevalent problem in such applications is that coefficients for individual vectors are
lost or erased. The redundancy of the semi-circular frame makes it quite robust to
such problems.
Here, we offer error bounds, presented in [4], on a computationally efficient scheme
for minimizing these errors. That is, with the loss of a single coefficient (or a family
of coefficients with no two consecutive) we can average the neighboring coefficients
to get perfect reconstruction. The endpoints are a special case which will be handled
later. The following lemma shows how this reconstruction scheme treats a single
missing coefficient, as long as the corresponding vector is not an “endpoint” vector
of the semicircular frame.
76
Lemma 8.2. Given a semi-circular frame, f ∈ Hd, with M vectors in each semi-
circle, and the coefficients for {fj}i−1j=1 and {fj}dMj=i+1, where i isn’t a multiple of M or
one more than a multiple of M ,
f =1
A
(i−1∑j=1
〈f, fj〉 fj +dM∑j=i+1
〈f, fj〉 fj +
⟨f,
fi−1 + fi+1
‖fi−1 + fi+1‖
⟩fi−1 + fi+1
‖fi−1 + fi+1‖
)
Also,
f =1
A
(i−2∑j=1
〈f, fj〉 fj +dM∑j=i+2
〈f, fj〉 fj +
(1 +
(1
cos2θ
)2)
(〈f, fi−1〉 fi−1 + 〈f, fi+1〉 fi+1)
),
where θ = πM.
The un-allowed indices correspond to the “endpoints” of the semicircles.
Proof. Since we know the arrangement of vectors in the semicircular frame, as long
as we don’t lose the vector coefficient on an endpoint of a semicircle, we can rebuild
a given vector by a linear combination of its immediate neighbors. Of course, the
aforementioned “endpoint” vectors only have one immediate neighbor in any given
semicircle. Since the vectors in a given semicircle have equal angles between them,
we know that the direction of the vector fi is fi−1 + fi+1. To normalize this direction,
we divide by the norm, ‖fi−1 + fi+1‖. It follows that
fi =fi−1 + fi+1
‖fi−1 + fi+1‖.
Substituting this expression in for fi yields the desired result.
The second equality is the same as the first, except that the norm of the vector
sum is given explicitly, with respect to M , and each vector is given with only one
coefficient.
77
In matrix notation, the compensation matrix for the reconstruction in Lemma 8.2
is:
C =
1 0 . . . . . . 0
0. . .
...... 1 1
‖fi−1+fi+1‖01
‖fi−1+fi+1‖ 1...
.... . . 0
0 . . . . . . 0 1
This leads us immediately to the following corollary, which basically says that we
can lose a lot of frame coefficients, as long as they are lost under some reasonable
assumptions.
Corollary 8.3. For every f ∈ Hd, f can be perfectly reconstructed with the frame
coefficients {fj}j∈J , where J is a subset of the integers between 1 and dM , such that
no two consecutive numbers are missing, and no missing number is a multiple of M
or one plus a multiple of M .
Remark 3. The above corollary shows just how efficient this reconstruction scheme
actually is. The corollary says we can loose nearly half the frame coefficients and
quickly get perfect reconstruction, while doing reconstruction by setting these coeffi-
cients equal to zero gives reconstruction error on the order of 1/2.
Now we have a handle on isolated erasures away from the endpoints, but what
of the endpoints? Depending on how the signal is reconstructed, this can result in
different kinds of error. First, let’s consider what happens if we leave out fi from
our sum altogether. Our error will be |〈f, fi〉|. Of course, it gets complicated if we
try to figure out how far away fi is from f , since the frame is so redundant. So we
78
could have a very small error if |〈f, fi〉| is close to zero, but if |〈f, fi〉| is one of the
largest coefficients, then the error is more substantial. However, in the latter case, the
vectors nearby would pick up some of the slack, as it were, of the missing coefficient.
The next most logical choice, which is putting say 〈f, fi+1〉 on fi in the sum, can
be sure to have a much smaller error.
∣∣∣∣∣∑j
〈f, fj〉fj −
(∑j 6=i
〈f, fj〉fj + 〈f, fi+1〉fi
)∣∣∣∣∣ = |〈f, fi〉 − 〈f, fi+1〉|, (8.1)
which is clearly minimized when f is in the hyperspace bisecting fi and fi+1. Call favg
the representative of that hyperspace in the plane formed by fi and fi+1, Therefore,
the error is maximized along the hyperspace perpendicular to favg. Call favg⊥ the
vector perpendicular to favg that also lies in the plane formed by fi and fi+1.
fff
f
θ2
i
i+1avg
avg
Figure 7: This shows favg and favg⊥ .
Our error can then be computed as
|〈favg⊥ , fi〉 − 〈favg⊥ , fi+1〉| = 2
(cos
(π − θ
2
)− cos
(π − θ
2
))= 4 cos
θ
2.
The next most logical choice is to just repeat an adjacent vector, in which case
the error is at most the difference between two adjacent vectors. In a similar manner
79
as the last case, if f is in the above mentioned hyperplane, the error is smaller than
if it is perpendicular to it, with respect to the origin. Here, the difference between
vectors has a magnitude of 2θπ−θ .
Of course, this leads us to ask what happens if we use the same algorithm as in
Lemma 8.2 when we are missing an endpoint. Not surprisingly, it turns out that
we could introduce some error. Compare fi (dotted vector) with f0 in the following
diagram. Here, θ is πM
, the angle between adjacent vectors in the M -semicircular
frame.
ff
i-1
i
i+1
f0
f
“z”
“x”
“y”
Figure 8: If fi is missing, we can construct f0 using fi−1 and fi+1.
For simplicity’s sake, let’s consider fi to be in the positive x direction, fi−1 to be
in the x− z plane, and fi+1 to be in the x− y plane. Let −→x be the unit vector in the
x direction. Note that this coincides with fi. Define −→y and −→z in a similar fashion.
We can say fi−1 = cos θ−→x + sin θ−→z and fi+1 = cos θ−→x + sin θ−→y . If we average fi−1
and fi+1, as in Lemma 8.2, we will get f0, as shown in the picture, to be
f0 =fi−1 + fi+1
‖fi−1 + fi+1‖=
2 cos θ−→x + sin θ−→z + sin θ−→y‖2 cos θ−→x + sin θ−→z + sin θ−→y ‖
.
So our error is
EC = |〈f, fi〉fi − 〈f, f0〉f0| =
80
= |〈f, fi〉fi −〈f, fi−1〉fi−1 + 〈f, fi+1〉fi−1 + 〈f, fi−1〉fi+1 + 〈f, fi+1〉fi+1|
‖2 cos θ−→x + sin θ−→z + sin θ−→y ‖=
=
((1− 4 cos θ2
2 + 2 cos θ2
)〈f, fi〉
)2
+2(
sin θ2〈f, fi−M+12〉+ sin θ2〈f, fi+M+1
2〉)2
2 + 2 cos θ2
12
So as long as we never lose two or more coefficients in a row, even in the worst case
scenario of losing all the endpoints, our error is only dEc. We could even be losing
around half of the total coefficients here! The expected value of our error is
E(error) =d∑i=0
(di
)iM−1EC .
8.3 Check matrices with entries from a finite field
Any type of system which has information traveling from a source to a sink may be
viewed as some type of code. When the pieces of information are discrete strings of
symbols of a fixed length from a particular alphabet, we are dealing with a particular
type of code called a block code. In a block code, we define a word to be any string of
symbols from the alphabet of the predetermined length. The set of all possible words
is called the codespace. There are a set number of codewords which are particular
words which are often assigned external meaning. The set of codewords is also referred
to as the code. Typically, there are many more words than codewords.
Under normal circumstances, a source will transmit only codewords. However,
since the medium through which the codewords are transmitted is often noisy, we
expect the sink to receive corrupted data regularly. If a code is designed well, and
the corruption is within expected bounds, the corrupted words will most likely not
become other codewords, and are therefore recognized as corrupted data. From here,
the system may or may not have some protocol in place for correcting errors. We do
81
not go into further detail about error correction, but recommend the free e-book [22]
for more information.
What we now consider is a special type of block code called a linear code. We will
assume that our alphabet is a finite field, Fq, and that our block length is d. Then
our codespace becomes Fdq . One essential object for a linear code to function is called
a check matrix. This is a matrix whose entries span the space of words which are not
codewords. Therefore, if a word has zero product with the check matrix, it must be
a codeword.
Suppose we are in a regime where we want to construct a linear code of length 2
from Fq, but we are not allowed to use all of F2q for some reason. Theorem 6.1 gives
us explicit bounds for when such a code is constructible, and Lemma 6.4 gives us an
example which shows that this bound is sharp.
Corollary 8.4. Given a set E ⊂ F2q, from which we wish to construct a length 2 linear
code, we are guaranteed the existence of a valid check matrix only when #E & q2.
82
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89
VITA
Steven Senger was born May 19, 1982 in North Kansas City, Missouri. In 2005
he graduated from University of Missouri with bachelor’s degrees in Electrical Engi-
neering, Computer Engineering, and Mathematics. In 2009, he obtained a master’s
degree in mathematics from the same university, where he plans to graduate with a
Ph.D in Mathematics. During his time at the university, he has recorded and played
live many types music, but mainly jazz and metal. He also has done an extensive
amount of rock climbing.
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