the existence of good extensible rank-1 lattices

http://www.elsevier.com/locate/jcoJournal of Complexity 19 (2003) 286–300

The existence of good extensible rank-1 lattices$

Fred J. Hickernella,* and Harald Niederreiterb

aDepartment of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, ChinabDepartment of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543,

Singapore

Received 5 April 2002; accepted 6 September 2002

Abstract

Extensible integration lattices have the attractive property that the number of points in the

node set may be increased while retaining the existing points. It is shown here that there exist

generating vectors, h; for extensible rank-1 lattices such that for n ¼ b; b2;y points and

dimensions s ¼ 1; 2;y the figures of merit Ra; Pa and discrepancy are all small. The upper

bounds obtained on these figures of merit for extensible lattices are some power of log n worse

than the best upper bounds for lattices where h is allowed to vary with n and s:r 2002 Elsevier Science (USA). All rights reserved.

Keywords: Discrepancy; Figures of merit; Extensible lattices; Lattice rules; Quasi-Monte Carlo

integration

1. Introduction

Multidimensional integrals over the unit cube are often approximated by takingthe average over a well-chosen set of pointsZ

½0;1Þsf ðxÞ dxE

1

n

Xn�1i¼0

f ðziÞ:

Quasi-Monte Carlo methods [9] choose the set fzig to be evenly distributed over theunit cube. One popular choice of points are the node sets of rank-1 integration

$This work was partially supported by a Hong Kong Research Grants Council Grant HKBU/2030/

99P.

*Corresponding author. Tel.: +1-852-3411-7015; fax: +1-852-3411-5811.

E-mail addresses: [email protected] (F.J. Hickernell), [email protected] (H. Niederreiter).

0885-064X/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0885-064X(02)00026-2

lattices (see [5]; [9, Chapter 5]; [12]). Such sets may be expressed as

ffih=n þ Dg: i ¼ 0; 1;y; n � 1g; ð1Þ

where h is an s-dimensional integer generating vector, D is an s-dimensional shift,and fg denotes the fractional part, i.e., fxg ¼ x ðmod 1Þ: The quality of the setdefined in (1) depends mainly on the choice of h and somewhat on the choice of D:There are several quality measures, including Ra; Pa; and the discrepancy, which willbe defined and discussed later in this article.One disadvantage of lattice rules for numerical integration has been that the

generating vector depends on n and s: If one changes either of these, then one mustalso change h: Recently, lattice node sets have been proposed that can be extended inn (for this reason they are called extensible lattices). For a fixed integer base bX2; let

any integer iX0 be expressed as i ¼ i1 þ i2b þ i3b2 þ?; where the digits ij are in

f0; 1;y; b � 1g: Then define the radical inverse function as

FbðiÞ ¼ i1b�1 þ i2b

�2 þ i3b�3 þ? :

The sequence

fzi ¼ fhFbðiÞ þ Dg: i ¼ 0; 1;yg

has the property that any finite piece with n ¼ bm points,

fzi ¼ fhFbði þ lbmÞ þ Dg ¼ fhFbðiÞ þ FbðlÞhb�m þ Dg: i ¼ 0; 1;y; bm � 1g;

l ¼ 0; 1;y; m ¼ 1; 2;y ð2Þ

is the node set of a shifted rank-1 lattice [2,3], just like the pieces of a ðt; sÞ-sequenceare ðt;m; sÞ-nets [9, Chapter 4]. Good generating vectors h that work simultaneouslyfor a range of m and s have been found experimentally [3]. The purpose of this articleis to prove that there exist generating vectors h; dependent on b but independent of m

and s; that give node sets of form (1) and (2) with figures of merit that are nearly asgood as the best known upper bounds for node sets where the generating vector isallowed to depend on m and s:The remainder of this introduction reviews several figures of merit for lattice rules.

The next section gives an upper bound on the value of Ra for extensible lattices. Thisis used to give upper bounds on Pa and discrepancy in the following two sections.

In this article it is assumed that the integrand, f ; is a function of x ¼ðx1; x2;yÞA½0; 1ÞN; and that the integration domain is ½0; 1ÞN: Integration

problems over ½0; 1Þs just assume that f depends only on the first s variables. Let1 : s denote the set f1;y; sg; and for any uC1 : N; let xu denote the vector indexedby the elements of u: Let juj denote the cardinality of u: Throughout this article it isassumed that u is a finite set.Suppose that the integrand can be written as an absolutely summable Fourier

series,

f ðxÞ ¼XkAZN

f ðkÞe2pikTx;

F.J. Hickernell, H. Niederreiter / Journal of Complexity 19 (2003) 286–300 287

where i ¼ffiffiffiffiffiffiffi�1

p: For any parameter cA½0;NÞN and kAZN let

r ðk; cÞ ¼YNj¼1

rðkj; gjÞ; where rðkj; gjÞ ¼1; kj ¼ 0;

g�1j jkjj; kja0:

(

For any a41; this quantity can be used to define a Banach space of functions:

Fa ¼ ffAL2ð½0; 1ÞNÞ: jj f jjFaoNg;

where

jj f jjFa¼ sup

kAZN

ðr ðk; cÞajf ðkÞjÞ:

If gj ¼ 0; then functions in Fa are assumed not to depend on the coordinate xj ; i.e.,

f ðkÞ ¼ 0 for all k with kja0: Let *Zu ¼ fkAZN: kj ¼ 0 8jeug: Then the subspace ofFa containing functions depending only on the coordinates indexed by u is Fu;a ¼ffAFa: f ðkÞ ¼ 0 8ke *Zug:For any positive integer n and any hAZN; the dual lattice consists of all wave

numbers k with kTh ¼ 0 mod n: The set Bðh; n; uÞ ¼ f0akA *Zu: kTh ¼ 0 mod ngconsists of all nonzero wave numbers in the dual lattice whose nonzero componentsare in the directions indexed by u: Let

Paðh; c; n; uÞ ¼X

kABðh;n;uÞr ðk; cÞ�a: ð3Þ

Then one may derive the following tight worst-case integration error bound forlattice rules using the node set (1) [1]:Z

½0;1ÞNf ðxÞ dx� 1

n

Xn�1i¼0

f ðziÞ��

��pPaðh; c; n; uÞjj f jjFa8fAFu;a: ð4Þ

For band-limited integrands one may derive a similar error bound. Let *Zun ¼

*Zu-ð�n=2; n=2 N and Fn;u;a ¼ ffAFu;a: f ðkÞ ¼ 0 8ke *Zung: Then the analogous

error bound to (4) isZ½0;1ÞN

f ðxÞ dx� 1

n

Xn�1i¼0

f ðziÞ��

��pRaðh; c; n; uÞjj f jjFa8fAFn;u;a; ð5Þ

where

Raðh; c; n; uÞ ¼X

kABðh;n;uÞr ðk; cÞ�a ð6Þ

and Bðh; n; uÞ ¼ Bðh; n; uÞ-ð�n=2; n=2 N: Note that whereas one must require a41for Paðh; c; n; uÞ to be defined and (4) to make sense, the error bound (5) is welldefined for aX0:Note also that both Paðh; c; n; uÞ and Raðh; c; n; uÞ do not depend onthe shift vector D:Lattice rules are not only used for periodic integrands, but also for non-periodic

ones. Let Wp denote the Banach space of functions that are absolutely continuous

on ½0; 1 N with Lp-integrable mixed partial derivatives of up to order one in each

F.J. Hickernell, H. Niederreiter / Journal of Complexity 19 (2003) 286–300288

coordinate direction. Let gu ¼Q

jAu gj: The norm for this space is defined as

jj f jjWp¼

XuD1:NjujoN

g�pu

Z½0;1 u

@jujf

@xu

��x1:NWu¼1

��p

dxu

8><>: g1=p; 1ppoN;

jj f jjWN¼ sup

uD1:NjujoN

g�1u supxuA½0;1 u

@jujf

@xu

��x1:NWu¼1

��:Again, if gj ¼ 0; then the functions inWp are assumed not to depend on xj : As above

let Wu;p denote the subspace of Wp whose elements do not depend on xj for jeu:

The integration error bound for this space is [4]Z½0;1ÞN

f ðxÞ dx� 1

n

Xn�1i¼0

f ðziÞ��

��pDn

qðfzig; c; n; uÞjj f jjWp

8fAWu;p;1

pþ 1

q¼ 1; ð7Þ

where the (weighted) Lq-star discrepancy is defined as

Dn

qðfzig; c; n; uÞ ¼X

|CvDu

gqv

Z½0;1 v

jdiscvð0; x; fzig; nÞjq dxv

( )1=q

;

1pqoN; ð8aÞ

Dn

Nðfzig; c; n; uÞ ¼ sup

|CvDu

gv supxvA½0;1 v

jdiscvð0; x; fzig; nÞj: ð8bÞ

The discrepancy function, discuðy; x; fzig; nÞ; measures the difference between thevolume of the box ½yu; xuÞ and the proportion of sample points inside it:

discuðy; x; fzig; nÞ ¼YjAu

ðxj � yjÞ �1

n

Xn�1i¼0

YjAu

ð1ðzij;NÞðxjÞ � 1ðzij

;NÞðyjÞÞ; ð8cÞ

where yjpxj for all jAu: Here 1fg denotes the characteristic function.

One may also define an Lq-unanchored discrepancy:

Dqðfzig; c; n; uÞ

¼X

|CvDu

2jvjgqv

Z½0;1 2v

yvpxv

jdiscvðy; x; fzig; nÞjq dyv dxv

8<:

9=;

1=q

; 1pqoN; ð9aÞ

DNðfzig; c; n; uÞ ¼ sup|CvDu

gv sup0pyvpxvp1

jdiscvðy; x; fzig; nÞj: ð9bÞ


Both the star and the unanchored discrepancies may be bounded as follows:

Dn

qðfzig; c; n; uÞ;Dqðfzig; c; n; uÞ

pX

|CvDu

gv sup0pyvpxvp1

jdiscvðy; x; fzig; nÞj; 1pqpN: ð10Þ

2. Existence of extensible lattices with small Ra

The existence of lattices with small Raðh; c; n; 1 : sÞ for fixed n and s has beenshown using averaging arguments. After computing an upper bound on the averageof Raðh; c; n; 1 : sÞ over some set of h; one argues that there exist at least some h withan Raðh; c; n; 1 : sÞ at least as small as that upper bound. In the following, anaveraging argument is also used, but with a difference. For extensible lattice rules itwill be argued that there exists some h; independent of n and s; such thatRaðh; c; n; 1 : sÞ is not too much worse than the upper bound on the average for all n

and s:For extensible lattices one needs to allow generating vectors, h; whose coordinates

are generalizations of integers. For any integer bX2 let Zb be the set of all b-adic

integers i ¼P

N

l¼1 ilbl�1; where ilAf0; 1;y; b � 1g for all lX1; see [7] for the theory

of b-adic integers. Note that Zb*Z: The set Zb contains too many bad numbers, sodefine

Hb ¼ fiAZb: gcdði1; bÞ ¼ 1g:

Then HN

b ¼ Hb � Hb �? is the set of candidates for h; anN-vector. The set Zb has

a probability measure such that the set of all iAZb with specified first l digits has

measure b�l : This probability measure conditional on Hb is denoted by mb: TheCartesian product HN

b has the product probability measure mNb : The following

upper bound on the average of R1ðh; 1; n; uÞ taken over HN

b is implied by the proof

of [8, Theorem 1]. For hAHN

b we define Raðh; c; n; uÞ with n ¼ b; b2;y in the

obvious way, namely by (6) with h considered modulo n:

Lemma 1. For any finite uC1 : N we haveZHN

b

R1ðh; 1; n; uÞ dmNb ðhÞpn�1ðb1 þ b2 log nÞjuj

for some positive absolute constants b1 and b2 and n ¼ b; b2;y :

Theorem 2. Suppose we are given a fixed integer bX2; a fixed cA½0;NÞN; a fixed

aX1; and a fixed e40:

(i) There exist a mNb -measurable GbCHN

b and some constant CRða; c; e; sÞ such that

for all hAHN

b WGb;


Raðh; c; n; 1 : sÞpCRða; c; e; sÞn�aðlog nÞaðsþ1Þ½log logðn þ 1Þ að1þeÞ;

n ¼ b; b2;y; s ¼ 1; 2;y : ð11Þ

Furthermore, one may make mNb ðGbÞ arbitrarily close to zero by choosing CRða; c; e; sÞlarge enough.

(ii) IfP

N

j¼1 gaj j½logð j þ 1Þ 1þeoN for some aA½1; a ; then for any fixed d40 there

exists some CRða; a; c; dÞ such that for all hAHN

b WGb we have

Raðh; c; n; 1 : sÞpCRða; a; c; dÞn�a=aþd; n ¼ b; b2;y; s ¼ 1; 2;y : ð12Þ

Again one may make mNb ðGbÞ arbitrarily close to zero by choosing the above leading

constant large enough.

(iii) IfP

N

j¼1 gaj oN for some aA½1; a ; then for any fixed d40 there exists a mNb -

measurable GbCHN

b such that for all hAHN

b WGb (12) is satisfied, but (11) may not

be. Here too, mNb ðGbÞ can be made arbitrarily close to zero.

Proof. Define the quantities

Raðh; c; n; sÞ ¼X

fsgDuD1:s

gauRaðh; 1; n; uÞ;


|CuD1:s

gauRaðh; 1; n; uÞ ¼Xs

d¼1Raðh; c; n; dÞ: ð13Þ

Note that Raðh; c; n; 1 : sÞpRaðh; c; n; sÞ: The proof here actually shows the above

conclusions for Raðh; c; n; sÞ rather than Raðh; c; n; 1 : sÞ because these are needed forTheorem 7.

There are two important relations among Raðh; c; n; sÞ for different parameters athat are used in this proof. Because Raðh; 1; n; uÞ is non-increasing for a increasing, itfollows that


|CuD1:s

gauRaðh; 1; n; uÞpX

|CuD1:s

gauRaðh; 1; n; uÞ

¼ Raðh; ca=a; n; sÞ ð14Þ

for aXa; where ca=a denotes the vector obtained by raising each component of c to

the power a=a: Furthermore, the definitions of Ra and Ra together with Jensen’sinequality imply that


|CuD1:s

gauRaðh; 1; n; uÞpX

|CuD1:s

½gauRaðh; 1; n; uÞ a=a

pX

|CuD1:s

gauRaðh; 1; n; uÞ

" #a=a

¼ ½Raðh; c; n; sÞ a=a ð15Þ

for aXa:


Lemma 1 implies the following upper bound on the average value of R1ðh; c; n; sÞ:ZHN

b

R1ðh; c; n; sÞ dmNb ðhÞpMðc; n; sÞ for n ¼ b; b2;y;

where

Mðc; n; sÞ :¼ n�1gsðb1 þ b2 log nÞYs�1j¼1

½1þ gjðb1 þ b2 log nÞ :

For any e40 and j ¼ 1; 2;y define cj :¼ cjðeÞ :¼ c0ðeÞj½logð j þ 1Þ 1þe; where c0ðeÞis chosen to be larger than

PN

j¼1 j�1½logð j þ 1Þ �1�e: A set of bad h for a particular

n ¼ bm and s is then defined as

Gbms ¼ fhAHN

b : R1ðh; c; bm; sÞ4cmcsMðc; bm; sÞg;

m ¼ 1; 2;y; s ¼ 1; 2;y :

These sets cannot be too large. In fact, mNb ðGbmsÞp1=ðcmcsÞ since

mNb ðGbmsÞcmcsMðc; bm; sÞpZ

Gbms

R1ðh; c; bm; sÞ dmNb ðhÞ

pZ

HN

b

R1ðh; c; bm; sÞmNb ðhÞpMðc; bm; sÞ:

The set of all bad h is defined as the union of these sets:

Gb ¼[Nm¼1

[Ns¼1

Gbms:

It follows that this set is also not too large, namely,

mNb ðGbÞpXNm¼1

XNs¼1

1

cmcs

¼ 1

c0ðeÞXNj¼1

1

j½logð j þ 1Þ 1þe

( )2

o1:

Thus, mNb ðHN

b WGbÞ40; i.e., there exist some h that are not in Gb: In fact,

mNb ðHN

b WGbÞ may be made arbitrarily close to 1 by choosing c0ðeÞ large enough.For any hAHN

b WGb one may derive the following upper bound on R1:

R1ðh; c; bm; sÞ ¼Xs

j¼1R1ðh; c; bm; jÞpcm

Xs

j¼1cjMðc; bm; jÞ

p c0ðeÞm½logðm þ 1Þ 1þeb�m

�Ys

j¼1½1þ gjc0ðeÞjflogð j þ 1Þg1þeðb1 þ b2m log bÞ

for m¼1; 2;y and s¼1; 2;y : Applying the inequality Raðh; c; n; sÞp½R1ðh; c; n; sÞ afrom (15) implies part (i) of Theorem 2.


IfP

N

j¼1 gaj j½logð j þ 1Þ 1þeoN; then by (14) above and Lemma 3 below it follows

that for any d40 and any hAHN

b WGb;

Raðh; c; bm; sÞpR1ðh; ca; bm; sÞpCRða; a; c; dÞbmð�1þdÞ;

m ¼ 1; 2;y; s ¼ 1; 2;y; aX1;

for some constant CRða; a; c; dÞ: Then applying (15) completes the proof of part (ii)of Theorem 2.

Lemma 1 also implies an upper bound on the average value of R1ðh; c; n; sÞ:ZHN

b

R1ðh; c; n; sÞ dmNb ðhÞpMðc; nÞ for n ¼ b; b2;y;

where it is assumed thatP

N

j¼1 gjoN and Mðc; nÞ is defined as

Mðc; nÞ :¼ n�1YNj¼1

½1þ gjðb1 þ b2 log nÞ :

With cm as above, a set of bad h for a particular n ¼ bm and s is then defined as

Gbms ¼ fhAHN

b : R1ðh; c; bm; sÞ4cmMðc; bmÞg; m ¼ 1; 2;y; s ¼ 1; 2;y :

By a similar argument to that above, mNb ðGbmsÞp1=cm for all m and s: Moreover,

since R1ðh; c; bm; sÞ is a non-decreasing function of s; it follows that Gbm1DGbm2D?:

Letting Gbm ¼S

N

s¼1 Gbms; it follows that mNb ðGbmÞp1=cm: If Gb ¼S

N

m¼1 Gbm; then

mNb ðGbÞo1 by the choice of the cm: Thus, for all hAHN

b WGb; it follows from (14)

and by substituting ca for c that

Raðh; c; bm; sÞp R1ðh; ca; bm; sÞpcmMðca; bmÞ

¼ c0ðeÞm½logðm þ 1Þ 1þeb�m

YNj¼1

½1þ gaj ðb1 þ b2m log bÞ

for all aX1; mX1; and sX1: Again using Lemma 3 and (15), it follows that ifPN

j¼1 gaj oN for some aA½1; a ; then for any fixed d40 there exists a CRða; a; c; dÞ

such that for all hAHN

b WGb;

Raðh; c; bm; sÞp½Raðh; c; bm; sÞ a=apCRða; a; c; dÞbmð�a=aþdÞ;

m ¼ 1; 2;y; s ¼ 1; 2;y :

This completes the proof of part (iii) of Theorem 2. &

The following lemma was used in the above proof. A similar result was proved in[4].


Lemma 3. Given a fixed *cA½0;NÞN; let

Sð*c;mÞ ¼YNj¼1

½1þ *gjm ; m ¼ 1; 2;y :

IfP

N

j¼1 *gjoN; then for any d40 it follows that Sð*c;mÞpCð*c; dÞbdm for m ¼ 1; 2;y :

Proof. Let sd ¼P

N

j¼dþ1 *gj; d ¼ 0; 1;y : Note that by increasing d one may make

sd arbitrarily small. We can assume w.l.o.g. that all sd40: Then applying somerelatively elementary inequalities yields

log½Sð*c;mÞ ¼XNj¼1

log½1þ *gjm

pXd

j¼1log½1þ s�1d þ *gjm þ

XNj¼dþ1

log½1þ *gjm

¼ d logð1þ s�1d Þ þXd

j¼1log½1þ *gjm=ð1þ s�1d Þ

þXN

j¼dþ1log½1þ *gjm

p d logð1þ s�1d Þ þ sdmXd

j¼1*gj þ m

XNj¼dþ1

*gj

p d logð1þ s�1d Þ þ sdðs0 þ 1Þm:

Thus, it follows that

Sð*c;mÞpð1þ s�1d Þdbmsd ðs0þ1Þ=log b:

Choosing d large enough to make sdpdðlog bÞ=ðs0 þ 1Þ completes the proof. &

Theorem 2 shows the existence of good extensible lattices with R1ðh; c; n; 1 : sÞ ¼Oðn�1ðlog nÞsþ1½log logðn þ 1Þ 1þeÞ for any s with n tending to infinity. This is slightlyworse than the result of [9, Theorem 5.10], which shows the existence of lattices with

R1 ¼ Oðn�1ðlog nÞsÞ for fixed s and n; and the lower bound of [6] of the same order.Thus, the price for an extensible lattice (in both s and n) is an extra factor of the

order ðlog nÞ½log logðn þ 1Þ 1þe:


3. Existence of extensible lattices with small Pa

Upper bounds on Pa follow from the upper bounds on Ra derived in the previoussection. The theorem below follows immediately from the following lemma, which isa minor generalization of [9, Theorem 5.5].

Lemma 4. For any integer nX2; finite set uC1 : N; cA½0;NÞN; a41; and h ¼ðhjÞNj¼1AZN with gcdðhj; nÞ ¼ 1 for all jAu we have

Paðh; c; n; uÞoRaðh; c; n; uÞ � 1þ exp 2zðaÞn�aXjAu

gaj

!

þ n�1exp 2zðaÞXjAu

gaj

!

� �1þ exp 2azðaÞn1�aXjAu

gaj =ð1þ 2gaj zðaÞÞ" #( )

;

where zðaÞ ¼P

N

k¼1 k�a is the Riemann zeta function.

Proof. The proof here follows that in [9, Theorem 5.5]. Starting from (3) thedefinition of Pa may be written as a sum of several pieces:

Paðh; c; n; uÞ ¼ � 1þXlA *Zu

r ðnl; cÞ�a þX

kABðh;n;uÞ

XlA *Zu

rðkþ nl; cÞ�a

¼ � 1þYjAu

XlAZ

rðnl; gjÞ�a

( )

þX

kABðh;n;uÞ

YjAu

XlAZ

rðkj þ nl; gjÞ�a

( ): ð16Þ

The second term in this equation may be simplified as

YjAu

XlAZ

rðnl; gjÞ�a

( )¼YjAu

½1þ 2gaj zðaÞn�a

¼ expXjAu

logð1þ 2gaj zðaÞn�aÞ" #

p exp 2zðaÞn�aXjAu

gaj

!:


The inner sum in the last term in (16) may be written as

XlAZ

rðkj þ nl; gjÞ�a ¼ rðkj; gjÞ�a þ gajXNl¼1

ðkj þ nlÞ�a þXNl¼1

ð�kj þ nlÞ�a

" #

p rðkj; gjÞ�a þ gajXNl¼1

ðnlÞ�a þXNl¼1

ð�n=2þ nlÞ�a

" #

¼ rðkj; gjÞ�a þ 2agaj n�aXNl¼1

ð2lÞ�a þXNl¼1

ð2l � 1Þ�a

" #

¼ rðkj; gjÞ�a þ 2agaj zðaÞn�a:

Thus, the last term in (16) satisfies

XkABðh;n;uÞ

YjAu

XlAZ


( )

pX

kABðh;n;uÞ

YjAu

frðkj; gjÞ�a þ 2agaj zðaÞn�ag

oRaðh; c; n; uÞ

þXvCu

YjAuWv

½2agaj zðaÞn�a XkA *Zu

n

kTh¼0 mod n

YjAv

rðkj; gjÞ�a

8>><>>:

9>>=>>;:

Using the same argument as in the proof of [9, Theorem 5.5], it follows that

XkA *Zu

n

kTh¼0 mod n

YjAv

rðkj; gjÞ�aonjuj�jvj�1YjAv

½1þ 2gaj zðaÞ :

Thus, the last term in (16) has the following upper bound:

XkABðh;n;uÞ

YjAu

XlAZ


( )

oRaðh; c; n; uÞ þ n�1XvCu

YjAuWv

½2agaj zðaÞn1�a YjAv

½1þ 2gaj zðaÞ ( )


¼ Raðh; c; n; uÞ � n�1YjAu

½1þ 2gaj zðaÞ

þ n�1YjAu

½1þ 2gaj zðaÞ þ 2agaj zðaÞn1�a

pRaðh; c; n; uÞ

þ n�1 exp 2zðaÞXjAu

gaj

!

� �1þ exp 2azðaÞn1�aXjAu

gaj =ð1þ 2gaj zðaÞÞ" #( )

:

Combining this bound with that for the second term in (16) completes the proof ofthis lemma. &

The lemma above implies that Paðh; c; n; uÞoRaðh; c; n; uÞ þ Oðn�aÞ; where theleading constant in the O term depends on u: This relationship is uniform in u ifP

N

j¼1 gaj oN: Thus, Theorem 2 and Lemma 4 imply the following existence theorem

for extensible lattices with good Pa:

Theorem 5. Theorem 2 also holds if one replaces Ra by Pa; allowing for a change of

constants as well.

For fixed s the quantity Paðh; n; 1 : sÞ is known to have the following lower andupper bounds if the generating vector is allowed to depend on n and s [10,11]:

C1ða; sÞn�aðlog nÞs�1pPaðh; n; 1 : sÞpC2ða; s; eÞn�aðlog nÞaðs�1Þþ1þe:

The lower bound holds for all h and the upper bound holds for suitable h: The priceof an extensible lattice is that the upper bound has now increased by roughly a factor

of ðlog nÞ2a�1: Eq. (11) applied to Pa improves upon [13, Theorem 2.1] because theright-hand side decays more quickly with n; and the same h works for all n ¼ bm aswell as all s: Eq. (12) applied to Pa improves upon [14, Theorem 3] because the sameh works for all n ¼ bm and all s:

4. Existence of extensible lattices with small discrepancy

The unanchored discrepancy defined in (9) is related to R1 [9, Chapter 5]. Thefollowing lemma, a slight generalization of [9, Theorems 3.10 and 5.6], makes thisrelationship explicit.

Lemma 6. For any integer n ¼ bm; m ¼ 1; 2;y; any finite subset uC1 : N; any

hAZN; and any corresponding node set of a shifted lattice, fzi ¼ fhFbðiÞ þ Dg: i ¼0; 1;y; bm � 1g; the unanchored LN-discrepancy of these points projected into the


coordinates indexed by u has the following upper bound:

sup0pyupxup1

jdiscuðy; x; fzig; nÞjp1� ð1� 1=nÞjuj þ 1

2R1ðh; 1; n; uÞ:

Proof. It suffices to note that the proof of [9, Theorem 3.10] works also for thediscrepancy extended over all intervals modulo 1 and that this discrepancy isinvariant under shifts of the node set modulo 1. &

Lemma 6 implies the following theorem for the existence of extensible lattice ruleswith small discrepancy.

Theorem 7. Suppose we are given a fixed integer bX2; a fixed cA½0;NÞN; and a fixed

e40:

(i) There exist a mNb -measurable GbCHN

b and some constant CDðc; e; sÞ such that for

all hAHN

b WGb and all DA½0; 1ÞN; the node set of the shifted lattice given in (2)

satisfies

Dn

qðfzig; c; n; 1 : sÞ;Dqðfzig; c; n; 1 : sÞ

pCDðc; e; sÞn�1ðlog nÞsþ1½log logðn þ 1Þ 1þe;

n ¼ b; b2;y; s ¼ 1; 2;y; 1pqpN: ð17Þ

Furthermore, one may make mNb ðGbÞ arbitrarily close to zero by choosing CDðc; e; sÞlarge enough.

(ii) IfP

N

j¼1 gj j½logð j þ 1Þ 1þeoN; then for any fixed d40 there exists some CDðc; dÞsuch that for all hAHN

b WGb and all DA½0; 1ÞN; the node set of the resulting shifted

lattice satisfies

Dn

qðfzig; c; n; 1 : sÞ;Dqðfzig; c; n; 1 : sÞpCDðc; dÞn�1þd;

n ¼ b; b2;y; s ¼ 1; 2;y; 1pqpN: ð18Þ

Again one may make mNb ðGbÞ arbitrarily close to zero by choosing the above leading

constant large enough.

(iii) IfP

N

j¼1 gjoN; then for any fixed d40 there exists a mNb -measurable GbCHN

b

such that for all hAHN

b WGb and all DA½0; 1ÞN (18) is satisfied, but (17) may not be.

Here too, mNb ðGbÞ can be made arbitrarily close to zero.


Proof. From (10) and Lemma 6 it follows that the star and unanchoreddiscrepancies have the following upper bound:

Dn


pX

|CuD1:s

gu sup0pyupxup1

jdiscuðy; x; fzig; nÞj

pX

|CuD1:s

gu 1� ð1� 1=nÞjuj þ 1

2R1ðh; 1; n; uÞ

� �

¼Ys

j¼1ð1þ gjÞ �

Ys

j¼1½1þ gjð1� 1=nÞ þ 1

2R1ðh; c; n; sÞ

¼Ys

j¼1ð1þ gjÞ 1�

Ys

j¼11�

gj

nð1þ gjÞ

!" #þ 1

2R1ðh; c; n; sÞ;

where R was defined in (13). Using the fact that logð1� xÞXxðlogð1� aÞÞ=a for0pxpao1; we obtain a lower bound on the second product:

logYs

j¼11�

gj

nð1þ gjÞ

! !¼Xs

j¼1log 1�

gj

nð1þ gjÞ

!

X logð1� 1=nÞXs

j¼1

gj

1þ gj

:

Thus,

Dn


pYs

j¼1ð1þ gjÞ 1� exp logð1� 1=nÞ

Xs

j¼1

gj

1þ gj

!" #þ 1

2R1ðh; c; n; sÞ

¼Ys

j¼1ð1þ gjÞ 1� 1� 1

n

� �Ps

j¼1 gj=ð1þgjÞ" #

þ 1

2R1ðh; c; n; sÞ:

Since the star and unanchored discrepancies are both bounded above by

R1ðh; c; n; sÞ=2 plus a term that is Oðn�1Þ uniformly in s assuming that the gj are

summable, the conclusions of this theorem follow immediately from the proof ofTheorem 2. &

Acknowledgments

The authors thank the participants in the 2001 Oberwolfach workshop onNumerical Integration and its Complexity for their helpful comments. We speciallythank Greg Wasilkowski for useful discussions.


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the existence of good extensible rank-1 lattices

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