davenport’s theorem in classical discrepancy theory

DAVENPORT’S THEOREM

IN CLASSICAL DISCREPANCY THEORY

William Chen

(Macquarie University)

Short presentation at

MCQMC2012

University of New South Wales

13 February 2012

Classical discrepancy problem formulated by Roth

10 2. THE CLASSICAL DISCREPANCY PROBLEM

Conjecture (van der Corput 1935). Suppose that (si)i∈N is a real sequencein I = [0, 1). Corresponding to any arbitrarily large real number κ, there exist apositive integer n and two subintervals I1 and I2, of equal length, of I such that

|Z(I1, n)− Z(I2, n)| > κ.

In short, this conjecture expresses the fact that no sequence can, in a certainsense, be too evenly distributed.

This conjecture is true, as shown by van Aardenne-Ehrenfest in 1945. Indeed,we have the following refinement.

Theorem 2.1 (van Aardenne-Ehrenfest 1949). Suppose that (si)i∈N is a realsequence in I = [0, 1). Suppose further that N ∈ N is sufficiently large. Then

(2.3) sup1!n!N0<α!1

|D([0, α), n)|# log log N

log log log N.

This result immediately raises the question of which functions f(N) satisfy thefollowing assertion.

Assertion A. For every real sequence (si)i∈N in I = [0, 1) and every N ∈ N,we have

(2.4) sup1!n!N0<α!1

|D([0, α), n)|# f(N).

Next, we consider Roth’s formulation of the problem in 1954.Suppose that P is a distribution of N points in the unit square [0, 1]2. For every

aligned rectangle B(x) = [0, x1)× [0, x2), where x = (x1, x2) ∈ [0, 1]2,

IRREGULARITIES OF POINT DISTRIBUTION

WILLIAM CHEN

Survey given at the Workshop onSmall Ball Inequalities in Analysis, Probability Theory, and Irregularities of Distribution,

American Institute of Mathematics,8 December 2008

x

B(x)

Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia

E-mail address: [email protected] Z[P;B(x)] denote the number of points of P that fall into B(x), and considerthe discrepancy

(2.5) D[P;B(x)] = Z[P;B(x)]−Nx1x2,

noting that Nx1x2 represents the expected number of points of P that fall into therectangle B(x).

We now consider the corresponding question of which functions g(N) satisfy thefollowing assertion.

Assertion B. For every distribution P of N points in the unit square [0, 1]2,we have

(2.6) supx∈[0,1]2

|D[P;B(x)]|# g(N).

P – set of N points in unit square [0,1]2

D[P;B(x)] = |P ∩B(x)| −Nµ(B(x))

↑ ↑actual point count expectation

‖D[P]‖2 =

(∫[0,1]2

|D[P;B(x)]|2 dx

)12

‖D[P]‖∞ = supx∈[0,1]2

|D[P;B(x)]|

Roth (1954): ∀N , ∀|P| = N , ‖D[P]‖2 � (logN)12

Davenport (1956): ∀N > 2, ∃|P| = N , ‖D[P]‖2 � (logN)12

Schmidt (1972): ∀N , ∀|P| = N , ‖D[P]‖∞ � logN

‖D[P]‖2 can be made to be substantially less than ‖D[P]‖∞

Roth (1954): ∀N , ∀|P| = N , ‖D[P]‖2 � (logN)12



‖D[P]‖2 can be made to be substantially less than ‖D[P]‖∞

Lev (1996): choose |P| = N with ‖D[P]‖2 � (logN)12

P + y – translation of P by y ∈ [0,1]2 modulo [0,1]2

supy∈[0,1]2

‖D[P + y]‖2 � ‖D[P]‖∞ � logN

the goodness of a distribution can be translated away

Roth (1954): ∀N , ∀|P| = N , ‖D[P]‖2 � (logN)12



C (1980): ∀q ∈ (0,∞), ∀N > 2, ∃|P| = N , ‖D[P]‖q �q (logN)12

‖D[P]‖q can be made to be substantially less than ‖D[P]‖∞

Bilyk, Lacey, Parrisis, Vagharshakyan (2009):

inf|P|=N

‖D[P]‖exp(Lα) � (logN)1−1α for 2 6 α <∞

estimate varies smoothly in range for α between (logN)12 and logN

many proofs of Davenport’s theorem

CHAPTER 5

Introduction to Upper Bounds

5.1. A Seemingly Trivial Argument

Let B denote a compact and convex set in the unit torus T2. For every realnumber λ ∈ [0, 1], every rotation θ ∈ [0, 2π] and every translation x ∈ T2, let

B(λ, θ,x) = {θ(λy) + x : y ∈ B}denote the similar copy of B obtained from B by a contraction by factor λ aboutthe origin, followed by an anticlockwise rotation by angle θ about the origin andthen by a translation by vector x. We denote by A(B) the collection of all similarcopies of B obtained this way.

We begin our discussion here by making an inadequate attempt to establish thefollowing variant of Theorem 3.4.

Theorem 5.1. Let B denote a compact and convex set in T2. For every naturalnumber N ! 2, there exists a distribution P of N points in T2 such that

supA∈A(B)

|D[P;A]|"B N14 (log N)

12 .

Such simple and perhaps naive attempts often play an important role in the studyof upper bounds. Remember that we need to find a good set of points, and we oftenstart by toying with some specific set of points which we hope will be good. Oftenit is not, but sometimes it permits us to bring in some stronger techniques at alater stage of the argument.

For simplicity, let us assume that the number of points is a perfect square, sothat N = M2 for some natural number M . We may then choose to split the unittorus T2 in the natural way into a union of N = M2 little squares of side lengthM−1, and then place a point in the centre of each little square.

! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! !

Suppose that A ∈ A(B) is a similar copy of a given fixed compact and convexset B. We now attempt to estimate the discrepancy D[P;A]. Let S denote thecollection of the N = M2 little squares S of side length M−1. The additive property

31

NO, it is fine to use a square lattice, but ...

at least try to rotate a suitably sized square lattice by a suitable angle

↑ ↑

fundamental region of area N−1 Hardy, Littlewood

Hardy, Littlewood (1920, 1922): lattice points in right-angled triangle

CHAPTER 7

Upper Bounds in the Classical Problem

7.1. Diophantine Approximation and Davenport Reflection

We begin by making a fatally flawed attempt to establish1 Theorem 2.10.Again, for simplicity, let us assume that the number of points is a perfect square,

so that N = M2 for some natural number M . We may then choose to split theunit square [0, 1]2 in the natural way into a union of N = M2 little squares ofsidelength M−1, and then place a point in the centre of each little square. Let Pbe the collection of these N = M2 points.

Let ξ be the second coordinate of one of the points of P. Clearly, there areprecisely M points in P sharing this second coordinate. Consider the discrepancy

(7.1) D[P;B(1, x2)]

of the rectangle B(1, x2) = [0, 1)× [0, x2). As x2 increases from just less than ξ tojust more than ξ, the value of (7.1) increases by M . It follows immediately that

‖D[P]‖∞ ! 12M # N

12 .

Let us make a digression to the work of Hardy and Littlewood on the distributionof lattice points in a right angled triangle. Consider a large right angled triangleT with two sides parallel to the coordinate axes. We are interested in the numberof points of the lattice Z2 that lie in T . For simplicity, the triangle T is placed sothat the horizontal side is precisely halfway between two neighbouring rows of Z2

and the vertical side is precisely halfway between two neighbouring columns of Z2.

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

!!!!!!!!!!!!!!!!

Note that the lattice Z2 has precisely one point per unit area, so we can think ofthe area of T as the expected number of lattice points in T . We therefore wishto understand the difference between the number of lattice points in T and thearea of T , and this is the discrepancy of Z2 in T . The careful placement of thehorizontal and vertical sides of T means that the discrepancy comes solely from thethird side of T . In the work of Hardy and Littlewood, it is shown that the size ofthe discrepancy when T is large is intimately related to the arithmetic properties

1It was put to the author by a rather preposterous engineering colleague many years ago that

this could be achieved easily by a square lattice in the obvious way. Not quite the case, as anobvious way would be far from so to this colleague.

41

good estimate when hypothenuse has slope badly approximable

rotate a suitably sized square lattice by a suitable angle

Lev’s result suggests that this may not be good enough

translate the lattice modulo a fundamental region and average

Beck, C (1997): this approach is good enough

• DIOPHANTINE APPROXIMATION + TRANSLATION

there is absolutely nothing new in this paper

diophantine approximation – Davenport’s idea

translation – Roth’s idea

Davenport (1956): M points in [0,1)× [0,M)

IRREGULARITIES OF POINT DISTRIBUTION

WILLIAM CHEN

Survey given at the Workshop onSmall Ball Inequalities in Analysis, Probability Theory, and Irregularities of Distribution,

American Institute of Mathematics,8 December 2008

(x, y)

B(x, y)

Department of Mathematics, Macquarie University, Sydney, NSW 2109, AustraliaE-mail address: [email protected]

Λ – lattice generated by (1,0) and (φ,1)

Q = Λ ∩ ([0,1)× [0,M)) of M points

E[Q;B(x, y)] = |Q ∩B(x, y)| − xy

Davenport (1956): M points in [0,1)× [0,M)

Λ – lattice generated by (1,0) and (φ,1)

Q = Λ ∩ ([0,1)× [0,M)) of M points

E[Q;B(x, y)] = |Q ∩B(x, y)| − xy

E[Q;B(x, y)] =∑

06=m∈Z

(1− e(−mx)

2πim

) ∑06n<y

e(φnm)

the term 1 arises from B(x, y) anchored at the origin

and causes technical difficulties

Λ′ – lattice generated by (1,0) and (−φ,1)

mirror image of Λ across vertical axis

Q′ = Λ′ ∩ ([0,1)× [0,M)) of M points

Q∗ = Q∪Q′ of 2M points

F [Q∗;B(x, y)] = |Q∗ ∩B(x, y)| − 2xy

F [Q∗;B(x, y)] =∑

06=m∈Z

(e(mx)− e(−mx)

2πim

) ∑06n<y

e(φnm)

• DIOPHANTINE APPROXIMATION + REFLECTION

Roth (1979):

Λ(t) = Λ + (t,0) where t ∈ [0,1]

horizontally translated lattice

Q(t) = Λ(t) ∩ ([0,1)× [0,M)) of M points

E[Q(t);B(x, y)] = |Q(t) ∩B(x, y)| − xy

E[Q(t);B(x, y)] =∑

06=m∈Z

(1− e(−mx)

2πim

) ∑06n<y

e(φnm)

e(tm)


summary:



van der Corput point set of N = 2h points

P2h = {(0.a1 . . . ah,0.ah . . . a1) : a1, . . . , ah ∈ {0,1}}

nice periodicity properties• If we only show [12 , 58 )× [0, 1), of area 1

8 , then there are 32× 18 = 4 points

of P5 in this rectangle, with vertical distance 14 apart.

• In fact, for any integers m and h satisfying 0 ≤ h ≤ s and 0 ≤ m < 2h,the rectangle [m2−h, (m+1)2−h)× [0, 1) contains 2s−h points of Ps, withvertical distance 2h−s apart.

• Any rectangle of the form [0, y1) × [0, y2) is contained in a union of atmost s+1 sets of the form [m2−h, (m+1)2−h)× [0, y2), where 0 ≤ h ≤ sand 0 ≤ m < 2h. Each such set has discrepancy less than 1, and so thediscrepancy of the set [0, y1) × [0, y2) is at most s + 1 # log N . Thisis the trivial estimate, obtained by Lerch in 1904 and is essentially bestpossible for the extreme discrepancy!

• (C + Skriganov) For every s ∈ N, the set Ps of 2s points satisfies∫[0,1]2

|D[Ps;B(y)]|2 dy = 2−6s2 + O(s),

and so does not give desired upper bound.

can assume that x1 ∈ 2−hZ, otherwise approximate

D[P2h;B(x1, x2)] =h∑∗i=1

(αi −Ψ

(x2 + βi

2i−h

))

sawtooth function Ψ(z) = z − [z]− 12

∗ – some terms are not present, summation depends on x1

functions Ψ form a quasi-orthogonal system with respect to x2

αi present because the rectangles are anchored at the origin

h∑∗i=1

h∑∗j=1

αiαj leads to ‖D[P2h]‖22 = 2−6h2 +O(h)

Halton, Zaremba (1969): P2h does not give Davenport’s theorem

consider the sawtooth function

7.4. ROTH’S PROBABILISTIC TECHNIQUE 11

interval [0, 2h), an interval of length equal to the period of the set Qh(t). Wetherefore need to study integrals of the form∫ 2h

0

ψ(2−s′(t− βs′))ψ(2−s′′(t− βs′′)) dt,

or when either or both of βs′ and βs′′ are replaced by γs′ and γs′′ respectively.

Lemma 7.7. Suppose that the integers s′ and s′′ satisfy 0 ! s′, s′′ ! h, and thatthe real numbers βs′ and βs′′ are fixed. Then∫ 2h

0

ψ(2−s′(t− βs′))ψ(2−s′′(t− βs′′)) dt = O(2h−|s′−s′′|).

Proof. The result is obvious if s′ = s′′. Without loss of generality, let us assumethat s′ > s′′. For every a = 0, 1, 2, . . . , 2s′−s′′ − 1, in view of periodicity, we have∫ 2h

0

ψ(2−s′(t− βs′))ψ(2−s′′(t− βs′′)) dt

=∫ 2h

0

ψ(2−s′(t + a2s′′ − βs′))ψ(2−s′′(t + a2s′′ − βs′′)) dt

=∫ 2h

0

ψ(2−s′(t + a2s′′ − βs′))ψ(2−s′′(t− βs′′)) dt,

with the last equality arising from the observation that

ψ(2−s′′(t + a2s′′ − βs′′)) = ψ(a + 2−s′′(t− βs′′)) = ψ(2−s′′(t− βs′′)).

2s′−s′′−1∑a=0

ψ(2−s′(t + a2s′′ − βs′)) = ψ(2−s′′(t− βs′))

at all points of continuity.

start with red one, translate by half a period to get blue one

now add them to get another sawtooth function with period halved

average of red and blue has half the magnitude from before

Roth (1980):

P2h(t) = P2h + (0, t) where t ∈ [0,1] modulo 1

vertical translation modulo 1

D[P2h(t);B(x1, x2)] =h∑∗i=1

(Ψ(zi + t

2i−h

)−Ψ

(wi + t

2i−h

))

a sum of quasi-orthogonal functions in t

• VAN DER CORPUT + TRANSLATION

Proinov (1988): reflection across a carefully chosen horizontal line

P ′2h

= {(p1,1− p2) : (p1, p2) ∈ P2h}• The following pictures shows Q5 superimposed on P5.

• We now face integrals of the type

I =∫ 1

0

φ

(y + yi

2−sLi

)φ

(y + yj

2−sLj

)dy + three other similar integrals

= O

((Li, Lj)2

LiLj

).

• Note that Ps ∪Qs is explicitly given.

Proinov (1988): reflection across a carefully chosen horizontal line

P ′2h

= {(p1,1− p2) : (p1, p2) ∈ P2h}

D[P2h;B(x1, x2)] =h∑∗i=1

(αi −Ψ

(x2 + βi

2i−h

))

D[P ′2h;B(x1, x2)] =h∑∗i=1

(−αi −Ψ

(x2 + γi

2i−h

))

D[P2h ∪ P′2h;B(x1, x2)] = −

h∑∗i=1

(Ψ(x2 + γi

2i−h

)+ Ψ

(x2 + βi

2i−h

))

a sum of quasi-orthogonal functions in x2

• VAN DER CORPUT + REFLECTION

summary:





question: is reflection or translation really necessary?

C, Skriganov (2002): van der Corput point set of n = ph points

Pph = {0.a1 . . . ah,0.ah . . . a1) : a1, . . . , ah ∈ {0,1, . . . , p− 1}}

⊕ – coordinatewise and digitwise addition modulo p

(Pph,⊕) group isomorphic to Zhp

group characters – base p Walsh functions, values p-th roots of unity

approximation of D[Pph;B(x)] expanded as Fourier–Walsh series

coefficients are orthogonal provided that p large enough, p > 11

• VAN DER CORPUT ALONE

what if p = 2 and no orthogonality of Fourier–Walsh coefficients?

digit shifts – C (1983)

hindsight explanation – digit shift is group isomorphic to Z2h2

∑t∈Z2h

2

Wl′(t)Wl′′(t) =

{4h if l′ = l′′

0 otherwise

backdoor orthogonality

foresight explanation

7.4. ROTH’S PROBABILISTIC TECHNIQUE 11

interval [0, 2h), an interval of length equal to the period of the set Qh(t). Wetherefore need to study integrals of the form∫ 2h

0

ψ(2−s′(t− βs′))ψ(2−s′′(t− βs′′)) dt,

or when either or both of βs′ and βs′′ are replaced by γs′ and γs′′ respectively.

Lemma 7.7. Suppose that the integers s′ and s′′ satisfy 0 ! s′, s′′ ! h, and thatthe real numbers βs′ and βs′′ are fixed. Then∫ 2h

0

ψ(2−s′(t− βs′))ψ(2−s′′(t− βs′′)) dt = O(2h−|s′−s′′|).

Proof. The result is obvious if s′ = s′′. Without loss of generality, let us assumethat s′ > s′′. For every a = 0, 1, 2, . . . , 2s′−s′′ − 1, in view of periodicity, we have∫ 2h

0

ψ(2−s′(t− βs′))ψ(2−s′′(t− βs′′)) dt

=∫ 2h

0

ψ(2−s′(t + a2s′′ − βs′))ψ(2−s′′(t + a2s′′ − βs′′)) dt

=∫ 2h

0

ψ(2−s′(t + a2s′′ − βs′))ψ(2−s′′(t− βs′′)) dt,

with the last equality arising from the observation that

ψ(2−s′′(t + a2s′′ − βs′′)) = ψ(a + 2−s′′(t− βs′′)) = ψ(2−s′′(t− βs′′)).

2s′−s′′−1∑a=0

ψ(2−s′(t + a2s′′ − βs′)) = ψ(2−s′′(t− βs′))

at all points of continuity.

foresight explanation• If we only show [12 , 58 )× [0, 1), of area 1

8 , then there are 32× 18 = 4 points

of P5 in this rectangle, with vertical distance 14 apart.

• In fact, for any integers m and h satisfying 0 ≤ h ≤ s and 0 ≤ m < 2h,the rectangle [m2−h, (m+1)2−h)× [0, 1) contains 2s−h points of Ps, withvertical distance 2h−s apart.

• Any rectangle of the form [0, y1) × [0, y2) is contained in a union of atmost s+1 sets of the form [m2−h, (m+1)2−h)× [0, y2), where 0 ≤ h ≤ sand 0 ≤ m < 2h. Each such set has discrepancy less than 1, and so thediscrepancy of the set [0, y1) × [0, y2) is at most s + 1 # log N . Thisis the trivial estimate, obtained by Lerch in 1904 and is essentially bestpossible for the extreme discrepancy!

• (C + Skriganov) For every s ∈ N, the set Ps of 2s points satisfies∫[0,1]2

|D[Ps;B(y)]|2 dy = 2−6s2 + O(s),

and so does not give desired upper bound.

take white strip, shift 3rd digit after decimal point of 1st coordinate

white strip moved distance 18 to the right

superimpose to get strip of width 18 with 8 periodic points

summary:





• VAN DER CORPUT + DIGIT SHIFT

• VAN DER CORPUT ALONE

• DIOPHANTINE APPROXIMATION ALONE – see Bilyk’s talk

davenport’s theorem in classical discrepancy theory

Documents