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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
From the Kentucky Sequence to Benford’sLaw through Zeckendorf Decompositions.
Steven J. Miller ([email protected])http://www.williams.edu/Mathematics/sjmiller/public_html
AMS Special Session on Difference EquationsGeorgetown University, Washington, DC, 3/7/15
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Introduction
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Collaborators and Thanks
Collaborators:Kentucky Sequence: Joint with Minerva Catral, Pari Ford,Pamela Harris & Dawn Nelson.Benfordness: Andrew Best, Patrick Dynes, Xixi Edelsbunner,Brian McDonald, Kimsy Tor, Caroline Turnage-Butterbaugh &Madeleine Weinstein.
Supported by:NSF Grants DMS1265673, DMS0970067, DMS1347804 andDMS0850577, AIM and Williams College.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
Zeckendorf’s TheoremEvery positive integer can be written uniquely as a sum ofnon-consecutive Fibonacci numbers.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
Zeckendorf’s TheoremEvery positive integer can be written uniquely as a sum ofnon-consecutive Fibonacci numbers.
Example: 51 =?
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
Zeckendorf’s TheoremEvery positive integer can be written uniquely as a sum ofnon-consecutive Fibonacci numbers.
Example: 51 = 34 + 17 = F8 + 17.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
Zeckendorf’s TheoremEvery positive integer can be written uniquely as a sum ofnon-consecutive Fibonacci numbers.
Example: 51 = 34 + 13 + 4 = F8 + F6 + 4.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
Zeckendorf’s TheoremEvery positive integer can be written uniquely as a sum ofnon-consecutive Fibonacci numbers.
Example: 51 = 34 + 13 + 3 + 1 = F8 + F6 + F3 + 1.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
Zeckendorf’s TheoremEvery positive integer can be written uniquely as a sum ofnon-consecutive Fibonacci numbers.
Example: 51 = 34 + 13 + 3 + 1 = F8 + F6 + F3 + F1.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Results
Fibonacci Numbers: Fn+1 = Fn + Fn−1;First few: 1,2,3,5,8,13,21,34,55,89, . . . .
Zeckendorf’s TheoremEvery positive integer can be written uniquely as a sum ofnon-consecutive Fibonacci numbers.
Example: 51 = 34 + 13 + 3 + 1 = F8 + F6 + F3 + F1.Example: 83 = 55 + 21 + 5 + 2 = F9 + F7 + F4 + F2.Observe: 51 miles ≈ 82.1 kilometers.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Old Results
Central Limit Type Theorem
As n → ∞, the distribution of number of summands inZeckendorf decomposition for m ∈ [Fn,Fn+1) is Gaussian.
500 520 540 560 580 600
0.005
0.010
0.015
0.020
0.025
0.030
Figure: Number of summands in [F2010,F2011); F2010 ≈ 10420.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Benford’s law
Definition of Benford’s LawA dataset is said to follow Benford’s Law (base B) if theprobability of observing a first digit of d is
logB
(
1 +1d
)
.
More generally probability a significant at most s is logB(s),where x = SB(x)10k with SB(x) ∈ [1,B) and k an integer.
Find base 10 about 30.1% of the time start with a 1, only4.5% start with a 9.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Previous Work
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Equivalent Definition of the Fibonaccis
Fibonaccis are the only sequence such that each integer canbe written uniquely as a sum of non-adjacent terms.
1,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Equivalent Definition of the Fibonaccis
Fibonaccis are the only sequence such that each integer canbe written uniquely as a sum of non-adjacent terms.
1, 2,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Equivalent Definition of the Fibonaccis
Fibonaccis are the only sequence such that each integer canbe written uniquely as a sum of non-adjacent terms.
1, 2, 3,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Equivalent Definition of the Fibonaccis
Fibonaccis are the only sequence such that each integer canbe written uniquely as a sum of non-adjacent terms.
1, 2, 3, 5,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Equivalent Definition of the Fibonaccis
Fibonaccis are the only sequence such that each integer canbe written uniquely as a sum of non-adjacent terms.
1, 2, 3, 5, 8,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Equivalent Definition of the Fibonaccis
Fibonaccis are the only sequence such that each integer canbe written uniquely as a sum of non-adjacent terms.
1, 2, 3, 5, 8, 13 . . . .
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Equivalent Definition of the Fibonaccis
Fibonaccis are the only sequence such that each integer canbe written uniquely as a sum of non-adjacent terms.
1, 2, 3, 5, 8, 13 . . . .
Key to entire analysis: Fn+1 = Fn + Fn−1.
View as bins of size 1, cannot use two adjacent bins:
[1] [2] [3] [5] [8] [13] · · · .
Goal: How does the notion of legal decomposition affectthe sequence and results?
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Generalizations
Generalizing from Fibonacci numbers to linearly recursivesequences with arbitrary nonnegative coefficients.
Hn+1 = c1Hn + c2Hn−1 + · · · + cLHn−L+1, n ≥ L
with H1 = 1, Hn+1 = c1Hn + c2Hn−1 + · · ·+ cnH1 + 1, n < L,coefficients ci ≥ 0; c1, cL > 0 if L ≥ 2; c1 > 1 if L = 1.
Zeckendorf: Every positive integer can be written uniquelyas
∑
aiHi with natural constraints on the ai ’s (e.g. cannotuse the recurrence relation to remove any summand).
Central Limit Type Theorem
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Example: the Special Case of L = 1, c1 = 10
Hn+1 = 10Hn, H1 = 1, Hn = 10n−1.
Legal decomposition is decimal expansion:∑m
i=1 aiHi :ai ∈ {0,1, . . . ,9} (1 ≤ i < m), am ∈ {1, . . . ,9}.
For N ∈ [Hn,Hn+1), first term is anHn = an10n−1.
Ai : the corresponding random variable of ai . The Ai ’s areindependent.
For large n, the contribution of An is immaterial.Ai (1 ≤ i < n) are identically distributed random variableswith mean 4.5 and variance 8.25.
Central Limit Theorem: A2 + A3 + · · · + An → Gaussianwith mean 4.5n + O(1) and variance 8.25n + O(1).
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequencewith Minerva Catral, Pari Ford, Pamela Harris & Dawn Nelson
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5, 8],
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5, 8], [11,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5, 8], [11, 16],
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5, 8], [11, 16], [21,
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5, 8], [11, 16], [21, 32], [43, 64], [85, 128].
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5, 8], [11, 16], [21, 32], [43, 64], [85, 128].
a2n = 2n and a2n+1 = 13(2
2+n − (−1)n):an+1 = an−1 + 2an−3,a1 = 1,a2 = 2,a3 = 3,a4 = 4.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Kentucky Sequence
Rule: (s,b)-Sequence: Bins of length b, and:
cannot take two elements from the same bin, and
if have an element from a bin, cannot take anything fromthe first s bins to the left or the first s to the right.
Fibonaccis: These are (s,b) = (1,1).
Kentucky: These are (s,b) = (1,2).
[1, 2], [3, 4], [5, 8], [11, 16], [21, 32], [43, 64], [85, 128].
a2n = 2n and a2n+1 = 13(2
2+n − (−1)n):an+1 = an−1 + 2an−3,a1 = 1,a2 = 2,a3 = 3,a4 = 4.
an+1 = an−1 + 2an−3: New as leading term 0.33
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
What’s in a name?
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
What’s in a name?
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Theorem: Gaussian Behavior
620 640 660 680 700 720
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Figure: Plot of the distribution of the number of summands for100,000 randomly chosen m ∈ [1, a4000) = [1, 22000) (so m has on theorder of 602 digits).
Proved Gaussian behavior.36
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Theorem: Geometric Decay for Gaps
Figure: Plot of the distribution of gaps for 10,000 randomly chosenm ∈ [1, a400) = [1, 2200) (so m has on the order of 60 digits).
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Theorem: Geometric Decay for Gaps
Figure: Plot of the distribution of gaps for 10,000 randomly chosenm ∈ [1, a400) = [1, 2200) (so m has on the order of 60 digits). Left(resp. right): ratio of adjacent even (resp odd) gap probabilities.
Again find geometric decay, but parity issues so break into evenand odd gaps.
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Other Rules(Coming Attractions)
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Tilings, Expanding Shapes
1
2
4 6
8
16
24
32
64 96
1, 2, 4, 6, 8, 16, 24, 32, 64, 96, ...
Figure: (left) Hexagonal tiling; (right) expanding triangle covering.
Theorem:A sequence uniquely exists, and similar to previous work candeduce results about the number of summands and thedistribution of gaps.
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Fractal Sets
Figure: Sierpinski tiling.41
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Upper Half Plane / Unit Disk
Figure: Plot of tesselation of the upper half plane (or unit disk) bythe fundamental domain of SL2(Z), where T sends z to z + 1 and Ssends z to −1/z.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Benfordness in IntervalJoint with Andrew Best, Patrick Dynes, Xixi Edelsbunner, Brian
McDonald, Kimsy Tor, Caroline Turnage-Butterbaugh andMadeleine Weinstein
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Benfordness in Interval
Theorem (SMALL 2014): Benfordness in Interval
The distribution of the summands in the Zeckendorfdecompositions, averaged over the entire interval [Fn,Fn+1),follows Benford’s Law.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Benfordness in Interval
Theorem (SMALL 2014): Benfordness in Interval
The distribution of the summands in the Zeckendorfdecompositions, averaged over the entire interval [Fn,Fn+1),follows Benford’s Law.
Example
Looking at the interval [F5,F6) = [8,13)
8 = 8 = F5
9 = 8 + 1 = F5 + F1
10 = 8 + 2 = F5 + F2
11 = 8 + 3 = F5 + F3
12 = 8 + 3 + 1 = F5 + F3 + F1
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Preliminaries for Proof
Density of S
For a subset S of the Fibonacci numbers, define the densityq(S,n) of S over the interval [1,Fn] by
q(S,n) =#{Fj ∈ S | 1 ≤ j ≤ n}
n.
Asymptotic Density
If limn→∞ q(S,n) exists, define the asymptotic density q(S) by
q(S) = limn→∞
q(S,n).
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Needed Input
Let Sd be the subset of the Fibonacci numbers which share afixed digit d where 1 ≤ d < B.
Theorem: Fibonacci Numbers Are Benford
q(Sd) = limn→∞
q(Sd ,n) = logB
(
1 +1d
)
.
Proof: Binet’s formula, Kronecker’s theorem on equidistributionof nα mod 1 for α 6∈ Q.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Random Variables
Random Variable from Decompositions
Let X (In) be a random variable whose values are the Fibonaccinumbers in [F1,Fn) and probabilities are how often they occurin decompositions of m ∈ In:
P (X (In) = Fk ) :=
Fk−1Fn−k−2µnFn−1
if 1 ≤ k ≤ n − 2
1µn
if k = n
0 otherwise,
where µn is the average number of summands in Zeckendorfdecompositions of integers in the interval [Fn,Fn+1).
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Approximations
Estimate for P (X (In) = Fk)
P (X (In) = Fk ) =1
µnφ√
5+ O
(
φ−2k + φ−2n+2k)
.
Constant Fringes Negligible
For any r (which may depend on n):
∑
r<k<n−r
P (X (In) = Fk ) = 1 − r · O(
1n
)
.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Estimating P (X (In) ∈ S)
Set r :=⌊
log nlogφ
⌋
.
Density of S over Zeckendorf Summands
We have
P (X (In) ∈ S) =nq(S)
µnφ√
5+ o(1) → q(s).
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Remark
Stronger result than Benfordness of Zeckendorfsummands.
Global property of the Fibonacci numbers can be carriedover locally into the Zeckendorf summands.
If we have a subset of the Fibonacci numbers S withasymptotic density q(S), then the density of the set S overthe Zeckendorf summands will converge to this asymptoticdensity.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Benfordness of Random and Zeckendorf DecompositionsJoint with Andrew Best, Patrick Dynes, Xixi Edelsbunner, Brian
McDonald, Kimsy Tor, Caroline Turnage-Butterbaugh andMadeleine Weinstein
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Random Decompositions
Theorem 2 (SMALL 2014): Random Decomposition
If we choose each Fibonacci number with probability q,disallowing the choice of two consecutive Fibonacci numbers,the resulting sequence follows Benford’s law.
Example: n = 10
F1 + F2 + F3 + F4 + F5 + F6 + F7 + F8 + F9 + F10
= 2 + 8 + 21 + 89
= 120
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Choosing a Random Decomposition
Select a random subset A of the Fibonaccis as follows:
Fix q ∈ (0,1).
Let A0 := ∅.
For n ≥ 1, if Fn−1 ∈ An−1, let An := An−1, else
An =
{
An−1 ∪ {Fn} with probability q
An−1 with probability 1 − q.
Let A :=⋃
n An.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Main Result
TheoremWith probability 1, A (chosen as before) is Benford.
Stronger claim: For any subset S of the Fibonaccis withdensity d in the Fibonaccis, S ∩ A has density d in A withprobability 1.
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Preliminaries
LemmaThe probability that Fk ∈ A is
pk =q
1 + q+ O(qk ).
Using elementary techniques, we get
LemmaDefine Xn := #An. Then
E [Xn] =nq
1 + q+ O(1)
Var(Xn) = O(n).
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Expected Value of Yn
Define Yn,S := #An ∩ S. Using standard techniques, we get
Lemma
E[Yn] =nqd
1 + q+ o(n).
Var(Yn,S) = o(n2).
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Expected Value of Yn
Define Yn,S := #An ∩ S. Using standard techniques, we get
Lemma
E[Yn] =nqd
1 + q+ o(n).
Var(Yn,S) = o(n2).
Immediately implies with probability 1 + o(1)
Yn,S =nqd
1 + q+ o(n), lim
n→∞
Yn,S
Xn= d .
Hence A ∩ S has density d in A, completing the proof.58
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
Zeckendorf Decompositions and Benford’s Law
Theorem (SMALL 2014): Benfordness of Decomposition
If we pick a random integer in [0,Fn+1), then with probability 1as n → ∞ its Zeckendorf decomposition converges to Benford’sLaw.
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Proof of Theorem
Choose integers randomly in [0,Fn+1) by randomdecomposition model from before.
Choose m = Fa1 + Fa2 + · · · + Faℓ∈ [0,Fn+1) with
probability
pm =
{
qℓ(1 − q)n−2ℓ if aℓ ≤ n
qℓ(1 − q)n−2ℓ+1 if aℓ = n.
Key idea: Choosing q = 1/ϕ2, the previous formulasimplifies to
pm =
{
ϕ−n if m ∈ [0,Fn)
ϕ−n−1 if m ∈ [Fn,Fn+1),
use earlier results.60
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
References
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
References
Beckwith, Bower, Gaudet, Insoft, Li, Miller and Tosteson: The Average GapDistribution for Generalized Zeckendorf Decompositions: The FibonacciQuarterly 51 (2013), 13–27.http://arxiv.org/abs/1208.5820
Best, Dynes, Edelsbrunner, McDonald, Miller, Turnage-Butterbaugh andWeinstein, Benford Behavior of Generalized Zeckendorf Decompositions, toappear in the Fibonacci Quarterly. http://arxiv.org/pdf/1409.0482
Best, Dynes, Edelsbrunner, McDonald, Miller, Turnage-Butterbaugh andWeinstein: Benford Behavior of Generalized Zeckendorf Decompositions,preprint. http://arxiv.org/pdf/1412.6839
Bower, Insoft, Li, Miller and Tosteson, The Distribution of Gaps betweenSummands in Generalized Zeckendorf Decompositions, preprint. http://arxiv.org/pdf/1402.3912
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Intro Previous Work Kentucky Sequence Other Rules Benfordness in Interval Random + Zeck Decomposition References
References
Catral, Ford, Harris, Miller and Nelson, Generalizing Zeckendorf’s Theorem: TheKentucky Sequence, submitted August 2014 to the Fibonacci Quarterly.http://arxiv.org/pdf/1409.0488.pdf
Kologlu, Kopp, Miller and Wang: On the number of summands in Zeckendorfdecompositions: Fibonacci Quarterly 49 (2011), no. 2, 116–130. http://arxiv.org/pdf/1008.3204
Miller and Wang: From Fibonacci numbers to Central Limit Type Theorems:Journal of Combinatorial Theory, Series A 119 (2012), no. 7, 1398–1413.http://arxiv.org/pdf/1008.3202
Miller and Wang: Survey: Gaussian Behavior in Generalized ZeckendorfDecompositions: Combinatorial and Additive Number Theory, CANT 2011 and2012 (Melvyn B. Nathanson, editor), Springer Proceedings in Mathematics &Statistics (2014), 159–173. http://arxiv.org/pdf/1107.2718
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