discrepancy bounds and related questions. dmitriy bilyk … · discrepancy bounds and related...

DISCREPANCY BOUNDS and RELATED QUESTIONS.

Dmitriy BilykUniversity of Minnesota

Equidistribution:Arithmetic, Computational and Probabilistic Aspects

Institute for Mathematical SciencesNational University of Singapore

May 3, 2019

Dmitriy Bilyk Discrepancy bounds & related questions

Uniformly distributed sequences

A sequence (xn) is uniformly distributed in [0, 1] iff

for any interval I ⊂ [0, 1] : limN→∞

#n ≤ N : xn ∈ IN

Discrepancy of a sequence

For a sequence ω = (ωn)∞n=1 and an interval I ⊂ [0, 1] considerthe quantity

∆N,I = ]ωn : ωn ∈ I;n ≤ N −N |I|.

DefineD∗N = sup

I⊂[0,1]

∣∣∆N,I

∣∣.A sequence (ωn)∞n=1 is u.d. in [0, 1] if and only if

limN→∞

D∗NN

#n ≤ N : xn ∈ IN

∆N,I = ]ωn : ωn ∈ I;n ≤ N −N |I|.

DefineD∗N = sup

I⊂[0,1]

∣∣∆N,I

∣∣.

A sequence (ωn)∞n=1 is u.d. in [0, 1] if and only if

limN→∞

D∗NN

#n ≤ N : xn ∈ IN

∆N,I = ]ωn : ωn ∈ I;n ≤ N −N |I|.

DefineD∗N = sup

I⊂[0,1]

∣∣∆N,I

∣∣.A sequence (ωn)∞n=1 is u.d. in [0, 1] if and only if

limN→∞

D∗NN

Discrepancy function

Consider a set PN ⊂ [0, 1]d consisting of N points:

Define the discrepancy function of the set PN as

DN (x) = ]PN ∩ [0, x) −Nx1x2 . . . xd

Extremal discrepancy (star-discrepancy):

‖DN‖∞ = supx∈[0,1]d

|DN (x)|.

Lp discrepancy: ‖DN‖p =

( ∫[0,1]d

|DN (x)|p dx)1/p

DN (x) = ]PN ∩ [0, x) −Nx1x2 . . . xd

‖DN‖∞ = supx∈[0,1]d

|DN (x)|.

( ∫[0,1]d

|DN (x)|p dx)1/p

DN (x) = ]PN ∩ [0, x) −Nx1x2 . . . xd

‖DN‖∞ = supx∈[0,1]d

|DN (x)|.

( ∫[0,1]d

|DN (x)|p dx)1/p

Equivalence of two formulations

Theorem (K. Roth, 1954)

The following are equivalent:

(i) For every ω = (ωn)∞n=1 ⊂ [0, 1],

D∗N (ω) & f(N)

for infinitely many values of N .

(ii) For any distribution PN ⊂ [0, 1]2 of N points,

‖DN‖∞ & f(N)

Equivalence of two formulations

Theorem (K. Roth, 1954)

The following are equivalent:

(i) For every ω = (ωn)∞n=1 ⊂ [0, 1]d−1,

D∗N (ω) & f(N)

for infinitely many values of N .

(ii) For any distribution PN ⊂ [0, 1]d of N points,

‖DN‖∞ & f(N)

Roth’s Theorem

Klaus Roth, October 29, 1925 – November 10, 2015

Theorem (ROTH, K. F. On irregularities of distribution,Mathematika 1 (1954), 73–79.)

There exists Cd ≥ 0 such that for any N -point set PN ⊂ [0, 1]d

‖DN‖2 ≥ Cd(logN)d−1

Roth’s Theorem

Klaus Roth, October 29, 1925 – November 10, 2015

‖DN‖∞ ≥ ‖DN‖2 ≥ Cd(logN)d−1

Irregularities of distribution: Roth’s Theorem

According to Roth himself, this was his favorite result.

William Chen (private communication)

Kenneth Stolarsky (private communication)

Ben Green (comment on Terry Tao’s blog)

Roth’s Theorem: legacy

‖DN‖2 ≥ Cd(logN)d−1

4 papers by Roth (On irregularities of distribution. I–IV)

10 papers by W.M. Schmidt (On irregularities ofdistribution. I–X)

2 by J. Beck (Note on irregularities of distribution. I–II)

4 by W. W. L. Chen (On irregularities of distribution.I–IV)

2 by Beck and Chen (Note on irregularities of distribution.I–II)

a book by Beck and Chen, “Irregularities of distribution”.

Roth’s theorem, extensions and sharpness

Theorem (Roth, 1954 (p = 2); Schmidt, 1977 (1 < p < 2))

The following estimate holds for all PN ⊂ [0, 1]d with#PN = N :

‖DN‖p & (logN)d−1

Theorem (Davenport, 1956 (d = 2, p = 2); Roth, 1979 (d ≥ 3,p = 2); Chen, 1982 (p > 2, d ≥ 3); Chen, Skriganov, 2000’s)

There exist sets PN ⊂ [0, 1]d with

‖DN‖p . (logN)d−1

Roth’s theorem, extensions and sharpness

Theorem (Roth, 1954 (p = 2); Schmidt, 1977 (1 < p < 2))

The following estimate holds for all PN ⊂ [0, 1]d with#PN = N :

‖DN‖p & (logN)d−1

Theorem (Davenport, 1956 (d = 2, p = 2); Roth, 1979 (d ≥ 3,p = 2); Chen, 1982 (p > 2, d ≥ 3); Chen, Skriganov, 2000’s)

There exist sets PN ⊂ [0, 1]d with

‖DN‖p . (logN)d−1

Roth’s orthogonal function method

Dyadic intervals in [0, 1]:

2n,m+ 1

): m,n ∈ Z, n ≥ 0, 0 ≤ m < 2n

L∞ normalized Haar function on a dyadic Interval I:hI = −1Ileft + 1Iright

Orthogonality:

〈hI′ , hI′′〉 =

0hI′(x)·hI′′(x) dx = 0, I ′, I ′′ ∈ D, I ′ 6= I ′′,

f ∈ L2([0, 1]) can be written as f =∑

I∈D∗〈f,hI〉|I| hI

2n,m+ 1

): m,n ∈ Z, n ≥ 0, 0 ≤ m < 2n

Orthogonality:

〈hI′ , hI′′〉 =

0hI′(x)·hI′′(x) dx = 0, I ′, I ′′ ∈ D, I ′ 6= I ′′,

2n,m+ 1

): m,n ∈ Z, n ≥ 0, 0 ≤ m < 2n

Orthogonality:

〈hI′ , hI′′〉 =

0hI′(x)·hI′′(x) dx = 0, I ′, I ′′ ∈ D, I ′ 6= I ′′,

2n,m+ 1

): m,n ∈ Z, n ≥ 0, 0 ≤ m < 2n

Orthogonality:

〈hI′ , hI′′〉 =

0hI′(x)·hI′′(x) dx = 0, I ′, I ′′ ∈ D, I ′ 6= I ′′,

For a dyadic rectangle R = I1 × ...× Id ⊂ [0, 1]2

hR(x) := hI1(x1) · ... · hId(xd)

f ∈ L2([0, 1]d): f =∑

R∈Dd∗

〈f,hR〉|R| hR

hR(x) := hI1(x1) · ... · hId(xd)

f ∈ L2([0, 1]d): f =∑

R∈Dd∗

〈f,hR〉|R| hR

hR(x) := hI1(x1) · ... · hId(xd)

f ∈ L2([0, 1]d): f =∑

R∈Dd∗

〈f,hR〉|R| hR

Main idea:

DN ≈∑

R: |R|≈ 1N

〈DN , hR〉|R|

Fix scale n such that N ≈ 2n

How many R such that |R| = 2−n? How many shapes?

The number of shapes is

#(r1, . . . , rd) ∈ Zd+ : r1+· · ·+rd = n =

(n+ d− 1

d− 1

)≈ nd−1

Orthogonality yields

‖DN‖2 & nd−1

2 ≈ (logN)d−1

Main idea:

DN ≈∑

R: |R|≈ 1N

〈DN , hR〉|R|

#(r1, . . . , rd) ∈ Zd+ : r1+· · ·+rd = n =

(n+ d− 1

d− 1

)≈ nd−1

‖DN‖2 & nd−1

2 ≈ (logN)d−1

Main idea:

DN ≈∑

R: |R|≈ 1N

〈DN , hR〉|R|

#(r1, . . . , rd) ∈ Zd+ : r1+· · ·+rd = n =

(n+ d− 1

d− 1

)≈ nd−1

‖DN‖2 & nd−1

2 ≈ (logN)d−1

Main idea:

DN ≈∑

R: |R|≈ 1N

〈DN , hR〉|R|

#(r1, . . . , rd) ∈ Zd+ : r1+· · ·+rd = n =

(n+ d− 1

d− 1

)≈ nd−1

‖DN‖2 & nd−1

2 ≈ (logN)d−1

Main idea:

DN ≈∑

R: |R|≈ 1N

〈DN , hR〉|R|

#(r1, . . . , rd) ∈ Zd+ : r1+· · ·+rd = n =

(n+ d− 1

d− 1

)≈ nd−1

‖DN‖2 & nd−1

2 ≈ (logN)d−1

L∞ estimates

Conjecture

‖DN‖∞ (logN)d−1

Theorem (Schmidt, 1972; Halasz, 1981)

In dimension d = 2 we have ‖DN‖∞ & logN

d = 2: Lerch, 1904; van der Corput, 1934

There exist PN ⊂ [0, 1]2 with ‖DN‖∞ ≈ logN

L∞ estimates

Conjecture

L∞ estimates

Conjecture

Low discrepancy sets

The van der Corput set with N = 212 points(0.x1x2...xn, 0.xnxn−1...x2x1

), xk = 0 or 1.

Discrepancy ≈ logN

The irrational (α =√

2) lattice with N = 212 points(n/N, nα

), n = 0, 1, ..., N − 1.

Discrepancy ≈ logN

Random set with N = 212 points

Discrepancy ≈√N

L∞ estimates

Conjecture

d ≥ 3, Halton, Hammersley (1960):

There exist PN ⊂ [0, 1]d with ‖DN‖∞ . (logN)d−1

L∞ estimates

Conjecture

d ≥ 3, Halton, Hammersley (1960):

There exist PN ⊂ [0, 1]d with ‖DN‖∞ . (logN)d−1

Conjectures and results

Conjecture 1

‖DN‖∞ & (logN)d−1

Conjecture 2

‖DN‖∞ & (logN)d2

“The Great Open Problem of Discrepancy Theory”(Beck, Chen)

Theorem (DB, M.Lacey, A.Vagharshakyan, 2008)

For d ≥ 3 there exists η = ηd > 0 such that

2+η .

Conjecture 1

Conjecture 2

2+η .

Conjecture 1

Conjecture 2

2+η .

Conjecture 1

Conjecture 2

2+η .

Conjectures: (logN)d−1 vs. (logN)d/2

Conjecture 1: (logN)d−1

classical conjecture;

the exponent d− 1 is natural;

the best known constructions(incl. recent lower bounds ofM. Levin);

estimates for smooth versionsof discrepancy;

Tusnady’s problem

(combinatorial discrepancy);

Conjecture 2: (logN)d/2

recent conjecture;

typical√

logN differencebetween expectation andsupremum of sums of Nsubgaussian variables;

Sharp for the linear harmonicanalysis model;

related conjectures in

probability and

approximation theory;

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

Tusnady’s problem

recent conjecture;

typical√

probability and

The small ball inequality

Instead of studying DN we shall look at∑

|R|=2−n

Small Ball Conjecture

For dimensions d ≥ 2, we have

nd−2

∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

d = 2: Talagrand, ’94; Temlyakov, ’95; DB, Feldheim ’15.

Sharpness: random signs/Gaussians.d−1

2 follows from an L2 estimate.

Connected to probability, approximation, discrepancy.

Known: d−12 − η(d) for d ≥ 3

(DB, Lacey, Vagharshakyan, 2008)

|R|=2−n

nd−2

∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

nd−2

∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

nd−2

∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

Sharpness: random signs/Gaussians.

d−12 follows from an L2 estimate.

|R|=2−n

nd−2

∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

nd−2

∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

nd−2

∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

The small ball conjecture and discrepancy

For dimensions d ≥ 2, we have for all choices of αR

(d−2)∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

Conjecture 2

In both conjectures one gains a square root over the L2

estimate.

(d−2)∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

Conjecture 2

estimate.

(d−2)∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

Conjecture 2

estimate.

Signed Small Ball Conjecture

For dimensions d ≥ 2, we have for all choices of αR = ±1∥∥∥ ∑|R|=2−n

∥∥∥∞

Conjecture 2

estimate.

Discrepancy estimates Small Ball inequality (signed)

Dimension d = 2

‖DN‖∞ & logN

∥∥∥∥ ∑|R|=2−n

∥∥∥∥∞

(Schmidt, ’72; Halasz, ’81) (Talagrand, ’94; Temlyakov, ’95)

Higher dimensions, L2 bounds

‖DN‖2 & (logN)(d−1)/2∥∥∥∥ ∑|R|=2−n

∥∥∥∥2

& n(d−1)/2

Higher dimensions, conjecture

‖DN‖∞ & (logN)d/2∥∥∥∥ ∑|R|=2−n

∥∥∥∥∞

& nd/2

Higher dimensions, known results

‖DN‖∞ & (logN)d−12 +η

∥∥∥∥ ∑|R|=2−n

∥∥∥∥∞

& nd−12 +η

Table: Discrepancy estimates and the signed Small Ball inequalityDmitriy Bilyk Discrepancy bounds & related questions

Riesz product

Small ball inequality (d=2)

For d = 2, we have∥∥∥ ∑|R|=2−n

∥∥∥∞

& 2−n∑

|R|=2−n

Riesz product: Ψ(x) =∏nk=1(1 + fk)

Sidon’s theorem

If a bounded 2π-periodic function f has lacunary Fourier series∞∑k=1

akeinkx, nk+1/nk > λ > 1, then∥∥f∥∥∞ &

∞∑k=1

Riesz product: PK(x) =∏Kk=1(1 + εk cosnkx)

Riesz product

∥∥∥∞

& 2−n∑

|R|=2−n

Sidon’s theorem

∞∑k=1

Riesz product

∥∥∥∞

& 2−n∑

|R|=2−n

Sidon’s theorem

∞∑k=1

Discrepancy function Lacunary Fourier series

DN (x) = #PN ∩ [0, x) −Nx1x2 f(x) ∼∑∞k=1 ck sinnkx,

nk> λ > 1

‖DN‖2 &√

logN ‖f‖2 ≡√∑

|ck|2(Roth, ’54)

‖DN‖∞ & logN ‖f‖∞ &∑|ck|

(Schmidt, ’72; Halasz, ’81) (Sidon, ’27)

Riesz product:∏(

1 + cfk)

Riesz product:∏(1 + cos(nkx+ φk)

)‖DN‖1 &

√logN ‖f‖1 & ‖f‖2

(Halasz, ’81) (Sidon, ’30)

Riesz product:∏(

1 + i · c√logN

Riesz product:∏(1 + i · |ck|‖f‖2 cos(nkx+ θk)

)Table: Discrepancy function and lacunary Fourier series

Connections between problems

Combinatorial discrepancy: Tusnady’s problem

Let Rd be the family of axis-parallel rectanglesP is a set of N points.Combinatorial discrepancy:

disc(N,Rd) = supP⊂[0,1)d

infχ:P→±1

supR∈Rd

∣∣∣∣ ∑p∈P∩R

∣∣∣∣

Transference Sos; Beck; Lovasz, Spencer, Vesztergombi; ...

inf ‖DN‖∞ . disc(N,Rd)

(logN)d−1 . disc(N,Rd) . (logN)d−12

Lower bound: Matousek, Nikolov, 2015

Upper bound: Nikolov, 2017

infχ:P→±1

supR∈Rd

∣∣∣∣ ∑p∈P∩R

∣∣∣∣Transference Sos; Beck; Lovasz, Spencer, Vesztergombi; ...

infχ:P→±1

supR∈Rd

∣∣∣∣ ∑p∈P∩R

infχ:P→±1

supR∈Rd

∣∣∣∣ ∑p∈P∩R

infχ:P→±1

supR∈Rd

∣∣∣∣ ∑p∈P∩R

Some lower and upper bounds in dimension d = 2

LOWER BOUND UPPER BOUND

Axis-parallel rectangles

L∞ logN logN

L2 log12 N log

Rotated rectangles

N1/4N1/4

√logN

Circles

N1/4N1/4

√logN

Convex Sets

N1/3N1/3 log4N

Higher dimensions: d ≥ 3

LOWER BOUND UPPER BOUND

Axis-parallel boxes

L∞ (logN)d−1

(logN)d−1

L2 (logN)d−1

2 (logN)d−1

Rotated boxes

N12− 1

2d N12− 1

N12− 1

2d N12− 1

Convex Sets

N1− 2d+1 N1− 2

d+1 logcN

Geometric discrepancy

No rotations: discrepancy ≈ logN

All rotations: discrepancy ≈ N1/4

(J. Beck, H. Montgomerry)

Partial rotations(lacunary sets, sets of small Minkowski dimension, etc)DB, X.Ma, C. Spencer, J. Pipher (2016)

Directional discrepancy

Let Ω ⊂ [0, π/2)

AΩ = rectangles pointing in directions of ΩDΩ(N) = inf

supR∈AΩ

|D(PN , R)|

Theorem (DB, Ma, Pipher, Spencer)

Lacunary directions, e.g. Ω = 2−n: DΩ(N) . log3N

Lacunary of order M , e.g.Ω = 2−n1 + 2−n2 + ... + 2−nM : DΩ(N) . logM+2N

“Superlacunary” sequence, e.g. 2−2n

DΩ(N) . logN · (log logN)2

Ω has small upper Minkowski dimension 0 ≤ d 1

DΩ(N) . Nd

d+1+ε

Let Ω ⊂ [0, π/2)AΩ = rectangles pointing in directions of Ω

DΩ(N) = infPN

supR∈AΩ

|D(PN , R)|

DΩ(N) . Nd

d+1+ε

DΩ(N) = infPN

supR∈AΩ

|D(PN , R)|

DΩ(N) . Nd

d+1+ε

DΩ(N) = infPN

supR∈AΩ

|D(PN , R)|

DΩ(N) . Nd

d+1+ε

DΩ(N) = infPN

supR∈AΩ

|D(PN , R)|

DΩ(N) . Nd

d+1+ε

DΩ(N) = infPN

supR∈AΩ

|D(PN , R)|

DΩ(N) . Nd

d+1+ε

discrepancy bounds and related questions. dmitriy bilyk … · discrepancy bounds and related...

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