Uniform distribution and quasi-Monte Carlo methods Johann ... · Geometric discrepancy No...

Discrepancy theory and harmonic analysis

Dmitriy BilykUniversity of Minnesota

“Uniform distribution and quasi-Monte Carlo methods”Johann Radon Institute for Computational and Applied

Mathematics (RICAM)Linz, Austria

October 14, 2013

Dmitriy Bilyk Discrepancy and analysis

Geometric discrepancy

No rotations: discrepancy ≈ (log N)α

(K. Roth, W. Schmidt, van der Corput)

All rotations: discrepancy ≈ Nβ

(J. Beck, H. Montgomerry)

Pictures taken from Matousek’s “Geometric Discrepancy: An Illustrated Guide”

Discrepancy bounds in dimension d = 2

LOWER BOUND UPPER BOUND

Axis-parallel rectangles

L∞ log N log N

L2 log12 N log

Rotated rectangles

N1/4N1/4√

Circles

N1/4N1/4√

Convex Sets

N1/3N1/3 log4 N

Higher dimensions: d ≥ 3

Axis-parallel boxes

L∞ (log N)d−1

(log N)d−1

L2 (log N)d−1

2 (log N)d−1

Rotated boxes

N12− 1

2d N12− 1

N12− 1

2d N12− 1

Convex Sets

N1− 2d+1 N1− 2

d+1 logc N

Discrepancy bounds in dimension d = 2

Axis-parallel rectangles

L∞ log N log N

L2 log12 N log

Rotated rectangles

N1/4N1/4√

Circles

N1/4N1/4√

Convex Sets

N1/3N1/3 log4 N

Discrepancy function Lacunary Fourier series

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

k=1 ck sin nkx ,nk+1

nk> λ > 1

‖DN‖2 &√

log N ‖f ‖2 ≡√∑

|ck |2(Roth, ’54)

‖DN‖∞ & log N ‖f ‖∞ &∑|ck |

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

Riesz product:∏(

1 + cfk)

Riesz product:∏(1 + cos(nkx + φk)

)‖DN‖1 &

√log N ‖f ‖1 & ‖f ‖2

(Halasz, ’81) (Sidon, ’30)

Riesz product:∏(

1 + i · c√log N

Riesz product:∏(1 + i · |ck |‖f ‖2

cos(nkx + θk))

Table : Discrepancy function and lacunary Fourier series

Sidon’s theorem and non-dense orbits

Sidon’s theorem:Let nj ⊂ N be lacunary with

nj> 1 + ε.

Then there exists θ ∈ [0, 1] such that∣∣∣∣∑j

cj sin(2πnjx)

∣∣∣∣ & ε| log ε|−1∑j

|cj |.

Peres–Schlag 2010:Let nj ⊂ N be lacunary with

nj> 1 + ε.

Then there exists θ ∈ [0, 1] such that for all j ∈ N

‖njθ‖ & ε| log ε|−1.

Sidon’s theorem and non-dense orbits

Sidon’s theorem:Let nj ⊂ N be lacunary with

nj> 1 + ε.

Then there exists θ ∈ [0, 1] such that∣∣∣∣∑j

cj sin(2πnjx)

∣∣∣∣ & ε| log ε|−1∑j

|cj |.

nj> 1 + ε.

Origins: Erdos’ question

nj> 1 + ε.

History:

Origins: Erdos’ question

nj> 1 + ε.

History:

Connections

Another question of Erdos (1987):Let S = nj ⊂ N be lacunary with

nj> 1 + ε.

Define the graph G on Z: (n,m) is an edge iff |n −m| ∈ S.– What is the chromatic number of G ?

Katznelson (2001):If ∃θ with inf ‖njθ‖ > δ, then chr(G ) < 1/δ.There exists a set A ⊂ N with upper densityd∗(A) & ε| log ε|−1 such that

(A− A) ∩ S = ∅.

(S = nj is not an intersective set).For H ⊂ N, define

δH = supd∗(A) : (A− A) ∩ H = ∅.

γH = infa0 : T (x) = a0 +∑n∈H

ah cos(2πnx) ≥ 0, T (0) = 1

Rusza (1984): γH ≥ δH .

Connections

nj> 1 + ε.

Define the graph G on Z: (n,m) is an edge iff |n −m| ∈ S.– What is the chromatic number of G ?Katznelson (2001):If ∃θ with inf ‖njθ‖ > δ, then chr(G ) < 1/δ.

There exists a set A ⊂ N with upper densityd∗(A) & ε| log ε|−1 such that

(A− A) ∩ S = ∅.

δH = supd∗(A) : (A− A) ∩ H = ∅.

γH = infa0 : T (x) = a0 +∑n∈H

ah cos(2πnx) ≥ 0, T (0) = 1

Rusza (1984): γH ≥ δH .

Connections

nj> 1 + ε.

Define the graph G on Z: (n,m) is an edge iff |n −m| ∈ S.– What is the chromatic number of G ?Katznelson (2001):If ∃θ with inf ‖njθ‖ > δ, then chr(G ) < 1/δ.There exists a set A ⊂ N with upper densityd∗(A) & ε| log ε|−1 such that

(A− A) ∩ S = ∅.

(S = nj is not an intersective set).

For H ⊂ N, define

δH = supd∗(A) : (A− A) ∩ H = ∅.

γH = infa0 : T (x) = a0 +∑n∈H

ah cos(2πnx) ≥ 0, T (0) = 1

Rusza (1984): γH ≥ δH .

Connections

nj> 1 + ε.

Define the graph G on Z: (n,m) is an edge iff |n −m| ∈ S.– What is the chromatic number of G ?Katznelson (2001):If ∃θ with inf ‖njθ‖ > δ, then chr(G ) < 1/δ.There exists a set A ⊂ N with upper densityd∗(A) & ε| log ε|−1 such that

(A− A) ∩ S = ∅.

δH = supd∗(A) : (A− A) ∩ H = ∅.

γH = infa0 : T (x) = a0 +∑n∈H

ah cos(2πnx) ≥ 0, T (0) = 1

Rusza (1984): γH ≥ δH .

Question

Given Ω ⊂ [0, 1], does there exists α such that∣∣∣∣(α− θ)− p

∣∣∣∣ > c

q2for all θ ∈ Ω ???

Marshall Hall (Annals of Math., 48, 1947):any real number is representable as a sum of two continuedfractions with partial quotients at most 4.

Given any θ1, θ2, there exists α such that∣∣∣∣(α− θ1)− p

∣∣∣∣ > c

∣∣∣∣(α− θ2)− p

∣∣∣∣ > c

Question

∣∣∣∣ > c

∣∣∣∣(α− θ2)− p

∣∣∣∣ > c

Question

∣∣∣∣ > c

∣∣∣∣(α− θ2)− p

∣∣∣∣ > c

Question

∣∣∣∣ > c

Lemma (Cassels (1956); Davenport (1964))

For all n > 0 there exists c = c(n) > 0 so that for any θ1, θ2, ..., θnthere exists α such that for all j = 1, ..., n∣∣∣∣

(α− θj)−p

∣∣∣∣ > c

for all p ∈ Z, q ∈ N.

c(n) ≈ 1/n2

Question

∣∣∣∣ > c

For all n > 0 there exists c = c(n) > 0 so that for any θ1, θ2, ..., θnthere exists α such that for all j = 1, ..., n∣∣∣∣

(α− θj)−p

∣∣∣∣ > c

c(n) ≈ 1/n2

Question

∣∣∣∣ > c

For all n > 0 there exists c = c(n) > 0 so that for any θ1, θ2, ..., θnthere exists α such that for all j = 1, ..., n∣∣∣∣ tan (α− θj)−

∣∣∣∣ > c

c(n) ≈ 1/n2

Question

∣∣∣∣ > c

For all n > 0 there exists c = c(n) > 0 so that for any θ1, θ2, ..., θnthere exists α such that for all j = 1, ..., n∣∣∣∣ tan (α− θj)−

∣∣∣∣ > c

for all p ∈ Z, q ∈ N. c(n) ≈ 1/n2

Question

∣∣∣∣ > c

q2 · ψ(q)for all θ ∈ Ω ???

Examples:

Lacunary sequences, e.g. 2−jLacunary sets of finite order:

Lacunary of order M + 1 = union of a set S lacunary of orderM and lacunary sequences converging to to each point of SDirectional maximal function MΩ is bounded in Lp(R2) iff Ω isa finite union of lacunary sets of finite order (Bateman, 2007)E.g., Ω = 2−n1 + 2−n2 + ... + 2−nMnk∈N

Sets of Minkowski dimension 0 ≤ d ≤ 1

“Superlacunary” sets, e.g. 2−2n.

Question

∣∣∣∣ > c

Examples:

Question

∣∣∣∣ > c

Examples:

Lacunary sequences, e.g. 2−j

Lacunary sets of finite order:

Question

∣∣∣∣ > c

Examples:

Question

∣∣∣∣ > c

Examples:

Question

∣∣∣∣ > c

Examples:

1. Cassels-Davenport iteration

Construct a sequence ...In−1 ⊃ In ⊃ ... with |In| → 0 so that∣∣∣∣(α− θ)− p

∣∣∣∣ ≥ c(n)

q2(∗)

for all α ∈ In, θ ∈ Ω, and R(n) ≤ q ≤ R(n + 1)—————————————————————————–

Let N(x) be the covering function of Ω (the minimal numberof intervals of length x needed to cover Ω).

Let F (x) ≈ x · N(x)

Theorem (DB, X. Ma, J. Pipher, C. Spencer)

There exists θ such that for all φ ∈ Ω∣∣∣∣(α− θ)− p

∣∣∣∣ & F−1

(F−1

))for all p ∈ Z, q ∈ N.

∣∣∣∣ ≥ c(n)

q2(∗)

∣∣∣∣ & F−1

(F−1

∣∣∣∣ ≥ c(n)

q2(∗)

∣∣∣∣ & F−1

(F−1

∣∣∣∣ ≥ c(n)

q2(∗)

∣∣∣∣ & F−1

(F−1

There exists α such that for all θ ∈ Ω∣∣∣∣(α− θ)− p

∣∣∣∣ & F−1

(F−1

Ω finite: N = #Ω, F (x) ≈ Nx ,

F−1(F−1(1/q2)) ≈ 1

Ω lacunary: N(x) ≈ log 1x , F (x) ≈ x · log 1

x , F−1(x) ≈ xlog(1/x) ,

F−1(F−1(1/q2)) ≈ 1

q2 log2 q

Ω “superlacunary”, F−1(F−1(1/q2)) ≈ 1q2(log log q)2

2. Trivial argument

Let h(q) be an increasing function of q such that∑ 1q·h(q) <∞. (h(q) = qε, h(q) = log1+ε q etc.)

For each q, cover Ω by N(δ(q)

)intervals Bk of length δ(q)

and remove the intervals Bk +(p/q − δ(q), p/q + δ(q)

Theorem

For any Ω ⊂ [0, 1] there exists α ∈ [0, 1) such that for all θ ∈ Ω∣∣∣∣(α− θ)− p

∣∣∣∣ & F−1

q2 · h(q)

Ω has upper Minkowski dimension d : N(x) .(

)d+ε,

F (x) . x1−d−ε, F−1(x) & x1

1−d−ε,

F−1(1/q2+ε) & q−2

1−d−ε

2. Trivial argument

Theorem

∣∣∣∣ & F−1

q2 · h(q)

)d+ε,

F (x) . x1−d−ε, F−1(x) & x1

1−d−ε,

F−1(1/q2+ε) & q−2

1−d−ε

2. Trivial argument

Theorem

∣∣∣∣ & F−1

q2 · h(q)

)d+ε,

F (x) . x1−d−ε, F−1(x) & x1

1−d−ε,

F−1(1/q2+ε) & q−2

1−d−ε

2. Trivial argument

Theorem

∣∣∣∣ & F−1

q2 · h(q)

)d+ε,

F (x) . x1−d−ε, F−1(x) & x1

1−d−ε,

F−1(1/q2+ε) & q−2

1−d−ε

3. Peres-Schlag method: more connections

nj> 1 + ε.

Littlewood conjecture: for any (α, β) ∈ R2,

infq≥1

q · ‖qα‖ · ‖qβ‖ = 0.

Moschevitin,... 2010-2012:e.g. Bugeaud–Moschevitin 2011:the set of points (α, β) such that inf q · log2 q‖qα‖ · ‖qβ‖ > 0has Hausdorff dimension 2.

Einsiedler, Katok, Lindenstrauss (2006):the set of points (α, β) such that inf q · ‖qα‖ · ‖qβ‖ > 0 hasHausdorff dimension 0.

nj> 1 + ε.

infq≥1

q · ‖qα‖ · ‖qβ‖ = 0.

nj> 1 + ε.

infq≥1

q · ‖qα‖ · ‖qβ‖ = 0.

nj> 1 + ε.

infq≥1

q · ‖qα‖ · ‖qβ‖ = 0.

3. Peres-Schlag method

Theorem (DB, Ma, Pipher, Spencer)

Assume that for every Q ∈ N we have

1/δ(Q)∑q=Q

q · F (δ(q)) . 1. Then

there exists α such that for all θ ∈ Ω∣∣∣∣(α− θ)− p

∣∣∣∣ & δ(q)

Ω lacunary of order M: N(x) ≈ logM 1x , F (x) ≈ x · logM 1

δ(q) ≈ 1

q2 logM+1 q

3. Peres-Schlag method

Assume that for every Q ∈ N we have

1/δ(Q)∑q=Q

q · F (δ(q)) . 1. Then

there exists α such that for all θ ∈ Ω∣∣∣∣(α− θ)− p

∣∣∣∣ & δ(q)

Ω lacunary of order M: N(x) ≈ logM 1x , F (x) ≈ x · logM 1

δ(q) ≈ 1

q2 logM+1 q

A. Erdos-Turan inequality

For a sequence ω = (ωn) ⊂ [0, 1]:

DN(ω) = supx⊂[0,1]

∣∣∣∣#n ≤ N : ωn ≤ x − Nx

∣∣∣∣

Theorem (Erdos-Turan)

For any sequence ω ⊂ [0, 1] we have

DN(ω) .N

m∑h=1

∣∣∣∣∣N∑

e2πihωn

∣∣∣∣∣for all natural numbers m.

For the Kronecker sequence ω = nα, we have∣∣∣∣∣N∑

e2πihnα

∣∣∣∣∣ ≤ 1

| sinπhα|.

‖hα‖.

DN(ω) = supx⊂[0,1]

∣∣∣∣#n ≤ N : ωn ≤ x − Nx

∣∣∣∣Theorem (Erdos-Turan)

DN(ω) .N

m∑h=1

∣∣∣∣∣N∑

e2πihωn

e2πihnα

∣∣∣∣∣ ≤ 1

| sinπhα|.

‖hα‖.

DN(ω) = supx⊂[0,1]

∣∣∣∣#n ≤ N : ωn ≤ x − Nx

∣∣∣∣Theorem (Erdos-Turan)

DN(ω) .N

m∑h=1

∣∣∣∣∣N∑

e2πihωn

e2πihnα

∣∣∣∣∣ ≤ 1

| sinπhα|.

‖hα‖.

For the sequence ω = nα we have

DN(ω) .N

m∑h=1

h · ‖hα‖

for all natural numbers m.

Assume that α satisfies∣∣∣α− p

∣∣∣ > 1q2·ψ(q)

for each p ∈ Z, q ∈ N.

We have

m∑h=1

h‖hα‖. ψ(2m) log m +

m∑h=1

ψ(2h) log h

m∑h=1

h‖hα‖. log2 m + ψ(m) +

m∑h=1

DN(ω) .N

m∑h=1

h · ‖hα‖

∣∣∣ > 1q2·ψ(q)

We have

m∑h=1

ψ(2h) log h

m∑h=1

DN(ω) .N

m∑h=1

h · ‖hα‖

∣∣∣ > 1q2·ψ(q)

We have

m∑h=1

ψ(2h) log h

m∑h=1

B. Partial quotients

For the Kronecker sequence ω = nα

DN(ω) .m+1∑j=1

where α = a0 + 1a1+ 1

a2+ 1...

= [a0; a1, a2, a3, ...],

= [a0; a1, a2, ..., an], and qm ≤ N < qm + 1.

If α satisfies∣∣∣α− p

∣∣∣ > 1q2·ψ(q) , then an+1 ≤ ψ(qn).

Ω “superlacunary”, e.g. Ω =

2−2m

F−1(F−1(1/q2)) ≈ 1q2(log log q)2 , i.e. ψ(q) = (log log q)2

One can deduce that

DN(ω) . log N · (log log N)2,

while Erdos-Turan would only yield log2 N.

DN(ω) .m+1∑j=1

where α = a0 + 1a1+ 1

a2+ 1...

= [a0; a1, a2, a3, ...],

= [a0; a1, a2, ..., an], and qm ≤ N < qm + 1.

∣∣∣ > 1q2·ψ(q) , then an+1 ≤ ψ(qn).

2−2m

One can deduce that

DN(ω) .m+1∑j=1

where α = a0 + 1a1+ 1

a2+ 1...

= [a0; a1, a2, a3, ...],

= [a0; a1, a2, ..., an], and qm ≤ N < qm + 1.

∣∣∣ > 1q2·ψ(q) , then an+1 ≤ ψ(qn).

2−2m

One can deduce that

DN(ω) .m+1∑j=1

where α = a0 + 1a1+ 1

a2+ 1...

= [a0; a1, a2, a3, ...],

= [a0; a1, a2, ..., an], and qm ≤ N < qm + 1.

∣∣∣ > 1q2·ψ(q) , then an+1 ≤ ψ(qn).

2−2m

One can deduce that

Directional Discrepancy: Setting

PN – a set of N points in [0, 1]d

R ⊂ [0, 1]d – a measurable set.

D(PN ,R) = ]PN ∩ R − N · vol (R)

Let Ω ⊂ [0, π/2)

AΩ = rectangles pointing in directions of Ω

DΩ(N) = infPN

supR∈AΩ

|D(PN ,R)|

All rotations

Ω = [0, π/2)

Theorem (J. Beck, 1987-88)

N1/4 . DΩ(N) . N1/4√

No rotations: axis-parallel rectangles

Ω = 0

Theorem (Lerch, 1904; Schmidt, 1972)

DΩ(N) ≈ log N

Theorem (L2: Roth, 1954; Davenport, 1956)

D(2)Ω (N) ≈

√log N

Ω finite −→ same (Beck-Chen, Chen-Travaglini)

No rotations: axis-parallel rectangles

Ω = 0

Theorem (Lerch, 1904; Schmidt, 1972)

DΩ(N) ≈ log N

Theorem (L2: Roth, 1954; Davenport, 1956)

D(2)Ω (N) ≈

√log N

Ω finite −→ same (Beck-Chen, Chen-Travaglini)

Irrational Lattice

Example

Let α ∈ [0, π]. Denote by PαN the lattice 1√NZ2 rotated by α and

intersected with [0, 1]2.

Theorem

If tanα is badly approximable, i.e.∣∣∣∣tan(α)− p

∣∣∣∣ & 1

for all p ∈ Z, q ∈ N, then the discrepancy of PαN with respect toaxis-parallel rectangles satisfies

D(PαN) ≈ log N.

Directional Discrepancy

DΩ(PαN ) . supθ∈Ω

Dc√N (n · tan(α− θ))

Lacunary directions: DΩ(N) . log3 N

Lacunary of order M: DΩ(N) . logM+2 N

“Superlacunary” sequence

DΩ(N) . log N · (log log N)2

Ω has upper Minkowski dimension 0 ≤ d < 1

DΩ(N) . Nd

d+1 +ε.

Directional Discrepancy

DΩ(PαN ) . supθ∈Ω

Dc√N (n · tan(α− θ))

Lacunary directions: DΩ(N) . log3 N

Lacunary of order M: DΩ(N) . logM+2 N

“Superlacunary” sequence

DΩ(N) . log N · (log log N)2

Ω has upper Minkowski dimension 0 ≤ d < 1

DΩ(N) . Nd

d+1 +ε.

Critical dimension

If Ω has upper Minkowski dimension 0 ≤ d ≤ 1:

DΩ(N) . Nd

d+1+ε. (∗)

but for Ω = [0, 2π), i.e. all rotations,

DΩ(N) . N1/4√

So (∗) is interesting only for dd+1 <

14 , i.e.

Critical dimension

DΩ(N) . Nd

d+1+ε. (∗)

DΩ(N) . N1/4√

14 , i.e.

Critical dimension

DΩ(N) . Nd

d+1+ε. (∗)

DΩ(N) . N1/4√

14 , i.e.

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