The Discrepancy Function and the Small Ball Inequality in Higher - - PowerPoint PPT Presentation

▶
the discrepancy function and the small ball inequality in
SMART_READER_LITE
LIVE PREVIEW

The Discrepancy Function and the Small Ball Inequality in Higher - - PowerPoint PPT Presentation

Mar 02, 2024 207 likes •1.22k views

The Discrepancy Function and the Small Ball Inequality in Higher Dimensions Dmitriy Bilyk Georgia Institute of Technology (joint work with M. Lacey and A. Vagharshakyan) 2007 Fall AMS Western Section Meeting University of New Mexico,

slide-1
SLIDE 1

The Discrepancy Function and the Small Ball Inequality in Higher Dimensions

Dmitriy Bilyk Georgia Institute of Technology (joint work with M. Lacey and A. Vagharshakyan) 2007 Fall AMS Western Section Meeting University of New Mexico, Albuquerque, NM October 14, 2007

Bilyk Discrepancy Function and the Small Ball Inequality

slide-2
SLIDE 2

Geometric Discrepancy

Let PN be an N point set in [0, 1]d of and let R ⊂ [0, 1]d be a rectangle with sides parallel to the axis.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-3
SLIDE 3

Geometric Discrepancy

Let PN be an N point set in [0, 1]d of and let R ⊂ [0, 1]d be a rectangle with sides parallel to the axis. D(PN, R) = ♯{PN ∩ R} − N · vol (R)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-4
SLIDE 4

Geometric Discrepancy

Let PN be an N point set in [0, 1]d of and let R ⊂ [0, 1]d be a rectangle with sides parallel to the axis. D(PN, R) = ♯{PN ∩ R} − N · vol (R) Star-discrepancy: D(PN) = sup

R

|D(PN, R)|

Bilyk Discrepancy Function and the Small Ball Inequality

slide-5
SLIDE 5

Geometric Discrepancy

Let PN be an N point set in [0, 1]d of and let R ⊂ [0, 1]d be a rectangle with sides parallel to the axis. D(PN, R) = ♯{PN ∩ R} − N · vol (R) Star-discrepancy: D(PN) = sup

R

|D(PN, R)| D(N) = inf

PN D(PN)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-6
SLIDE 6

Geometric Discrepancy

Let PN be an N point set in [0, 1]d of and let R ⊂ [0, 1]d be a rectangle with sides parallel to the axis. D(PN, R) = ♯{PN ∩ R} − N · vol (R) Star-discrepancy: D(PN) = sup

R

|D(PN, R)| D(N) = inf

PN D(PN)

Can D(N) be bounded?

Bilyk Discrepancy Function and the Small Ball Inequality

slide-7
SLIDE 7

Geometric Discrepancy

Let PN be an N point set in [0, 1]d of and let R ⊂ [0, 1]d be a rectangle with sides parallel to the axis. D(PN, R) = ♯{PN ∩ R} − N · vol (R) Star-discrepancy: D(PN) = sup

R

|D(PN, R)| D(N) = inf

PN D(PN)

Can D(N) be bounded? NO (van Aardenne-Ehrenfest; Roth)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-8
SLIDE 8

Discrepancy function

Enough to consider rectangles with a vertex at the origin

Bilyk Discrepancy Function and the Small Ball Inequality

slide-9
SLIDE 9

Discrepancy function

Enough to consider rectangles with a vertex at the origin DN(x) = ♯{PN ∩ [0, x)} − Nx1x2 . . . xd

Bilyk Discrepancy Function and the Small Ball Inequality

slide-10
SLIDE 10

Discrepancy function

Enough to consider rectangles with a vertex at the origin DN(x) = ♯{PN ∩ [0, x)} − Nx1x2 . . . xd Lower estimates for DNp ???

Bilyk Discrepancy Function and the Small Ball Inequality

slide-11
SLIDE 11

Koksma-Hlawka Inequality

  • [0,1]d f (x) dx − 1

N

  • p∈PN

f (p)

  • 1

N V (f ) · DN∞

Bilyk Discrepancy Function and the Small Ball Inequality

slide-12
SLIDE 12

Koksma-Hlawka Inequality

  • [0,1]d f (x) dx − 1

N

  • p∈PN

f (p)

  • 1

N V (f ) · DN∞ V (f ) is the Hardy-Krause variation of f

Bilyk Discrepancy Function and the Small Ball Inequality

slide-13
SLIDE 13

Koksma-Hlawka Inequality

  • [0,1]d f (x) dx − 1

N

  • p∈PN

f (p)

  • 1

N V (f ) · DN∞ V (f ) is the Hardy-Krause variation of f V (f ) ” = ”

  • [0,1]d
  • ∂df

∂x1∂x2...∂xd

  • dx1 . . . dxd

Bilyk Discrepancy Function and the Small Ball Inequality

slide-14
SLIDE 14

An example

Consider a √ N × √ N lattice

Bilyk Discrepancy Function and the Small Ball Inequality

slide-15
SLIDE 15

An example

Consider a √ N × √ N lattice

Bilyk Discrepancy Function and the Small Ball Inequality

slide-16
SLIDE 16

An example

Consider a √ N × √ N lattice |DN(P) − DN(Q)| ≈ N ·

1 √ N =

√ N

Bilyk Discrepancy Function and the Small Ball Inequality

slide-17
SLIDE 17

An example

Consider a √ N × √ N lattice |DN(P) − DN(Q)| ≈ N ·

1 √ N =

√ N Thus DN∞ N

1 2 Bilyk Discrepancy Function and the Small Ball Inequality

slide-18
SLIDE 18

Lp estimates, 1 < p < ∞

Theorem DNp (log N)

d−1 2

Roth (p=2), Schmidt

Bilyk Discrepancy Function and the Small Ball Inequality

slide-19
SLIDE 19

Lp estimates, 1 < p < ∞

Theorem DNp (log N)

d−1 2

Roth (p=2), Schmidt Theorem There exist PN ⊂ [0, 1]d with DNp ≈ (log N)

d−1 2

(Davenport, Roth, Frolov, Chen)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-20
SLIDE 20

L∞ estimates

Conjecture DN∞ ≫ (log N)

d−1 2 Bilyk Discrepancy Function and the Small Ball Inequality

slide-21
SLIDE 21

L∞ estimates

Conjecture DN∞ ≫ (log N)

d−1 2

Theorem (Schmidt) For d = 2 we have DN∞ log N

Bilyk Discrepancy Function and the Small Ball Inequality

slide-22
SLIDE 22

L∞ estimates

Conjecture DN∞ ≫ (log N)

d−1 2

Theorem (Schmidt) For d = 2 we have DN∞ log N d = 2 van der Corput: There exist PN ⊂ [0, 1]2 with DN∞ ≈ log N

Bilyk Discrepancy Function and the Small Ball Inequality

slide-23
SLIDE 23

L∞ estimates

Conjecture DN∞ ≫ (log N)

d−1 2

Theorem (Schmidt) For d = 2 we have DN∞ log N d = 2 van der Corput: There exist PN ⊂ [0, 1]2 with DN∞ ≈ log N d ≥ 3 Halasz: There exist PN ⊂ [0, 1]d with DN∞ ≈ (log N)d−1

Bilyk Discrepancy Function and the Small Ball Inequality

slide-24
SLIDE 24

Conjectures

Conjecture 1 DN∞ (log N)d−1

Bilyk Discrepancy Function and the Small Ball Inequality

slide-25
SLIDE 25

Conjectures

Conjecture 1 DN∞ (log N)d−1 Conjecture 2 DN∞ (log N)

d 2 Bilyk Discrepancy Function and the Small Ball Inequality

slide-26
SLIDE 26

Roth’s Orthogonal Function Method: Definitions

Dyadic intervals are intervals of the form [k2q, (k + 1)2q).

Bilyk Discrepancy Function and the Small Ball Inequality

slide-27
SLIDE 27

Roth’s Orthogonal Function Method: Definitions

Dyadic intervals are intervals of the form [k2q, (k + 1)2q). For a dyadic interval I: hI = −1Ileft + 1Iright, Haar functions with L∞ normalization.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-28
SLIDE 28

Roth’s Orthogonal Function Method: Definitions

Dyadic intervals are intervals of the form [k2q, (k + 1)2q). For a dyadic interval I: hI = −1Ileft + 1Iright, Haar functions with L∞ normalization. In higher dimensions: for a rectangle R = I1 × · · · × Id hR(x1, . . . , xd) := hI1(x1) · ... · hId(xd)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-29
SLIDE 29

Roth’s Orthogonal Function Method: Definitions

All rectangles are in the unit cube in Rd.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-30
SLIDE 30

Roth’s Orthogonal Function Method: Definitions

All rectangles are in the unit cube in Rd. Choose n appropriately so that n ≈ log2 N and consider dyadic rectangles of volume 2−n.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-31
SLIDE 31

Roth’s Orthogonal Function Method: Definitions

All rectangles are in the unit cube in Rd. Choose n appropriately so that n ≈ log2 N and consider dyadic rectangles of volume 2−n. Hd

n = {(r1, r2, . . . , rd) ∈ Nd : r1 + · · · + rd = n}

Bilyk Discrepancy Function and the Small Ball Inequality

slide-32
SLIDE 32

Roth’s Orthogonal Function Method: Definitions

All rectangles are in the unit cube in Rd. Choose n appropriately so that n ≈ log2 N and consider dyadic rectangles of volume 2−n. Hd

n = {(r1, r2, . . . , rd) ∈ Nd : r1 + · · · + rd = n}

Definition For r ∈ Hd

n, call f an

r function iff it is of the form f =

  • R : |Rj|=2−rj

εR hR , εR ∈ {±1}.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-33
SLIDE 33

Roth’s Orthogonal Function Method: Proof

For each r ∈ Hd

n, there exists fr such that DN, fr 1.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-34
SLIDE 34

Roth’s Orthogonal Function Method: Proof

For each r ∈ Hd

n, there exists fr such that DN, fr 1.

Construct the test function F =

  • r∈Hd

n fr. Bilyk Discrepancy Function and the Small Ball Inequality

slide-35
SLIDE 35

Roth’s Orthogonal Function Method: Proof

For each r ∈ Hd

n, there exists fr such that DN, fr 1.

Construct the test function F =

  • r∈Hd

n fr.

♯{ r ∈ Hd

n} ≈ nd−1 ≈ (log N)d−1.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-36
SLIDE 36

Roth’s Orthogonal Function Method: Proof

For each r ∈ Hd

n, there exists fr such that DN, fr 1.

Construct the test function F =

  • r∈Hd

n fr.

♯{ r ∈ Hd

n} ≈ nd−1 ≈ (log N)d−1.

F2 ≈ (log N)

d−1 2 . Bilyk Discrepancy Function and the Small Ball Inequality

slide-37
SLIDE 37

Roth’s Orthogonal Function Method: Proof

For each r ∈ Hd

n, there exists fr such that DN, fr 1.

Construct the test function F =

  • r∈Hd

n fr.

♯{ r ∈ Hd

n} ≈ nd−1 ≈ (log N)d−1.

F2 ≈ (log N)

d−1 2 .

DN, F (log N)d−1.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-38
SLIDE 38

Roth’s Orthogonal Function Method: Proof

For each r ∈ Hd

n, there exists fr such that DN, fr 1.

Construct the test function F =

  • r∈Hd

n fr.

♯{ r ∈ Hd

n} ≈ nd−1 ≈ (log N)d−1.

F2 ≈ (log N)

d−1 2 .

DN, F (log N)d−1. Thus DN2 ≥ DN, fr F2 (log N)

d−1 2 Bilyk Discrepancy Function and the Small Ball Inequality

slide-39
SLIDE 39

The Small Ball Inequality

We are interested in non-trivial lower bounds for the ’hyperbolic’ sums of Haar functions:

  • |R|=2−n

αR hR

  • ∞

Bilyk Discrepancy Function and the Small Ball Inequality

slide-40
SLIDE 40

The ‘Trivial Bound’

L2 estimate n

d−1 2

  • |R|=2−n

αR hR

  • ∞ 2−n
  • |R|=2−n

|αR|

Bilyk Discrepancy Function and the Small Ball Inequality

slide-41
SLIDE 41

The ‘Trivial Bound’

L2 estimate n

d−1 2

  • |R|=2−n

αR hR

  • ∞ 2−n
  • |R|=2−n

|αR| holds with the L2 norm on the left hand side

Bilyk Discrepancy Function and the Small Ball Inequality

slide-42
SLIDE 42

The ‘Trivial Bound’

L2 estimate n

d−1 2

  • |R|=2−n

αR hR

  • ∞ 2−n
  • |R|=2−n

|αR| holds with the L2 norm on the left hand side Cauchy-Schwarz inequality

Bilyk Discrepancy Function and the Small Ball Inequality

slide-43
SLIDE 43

The ‘Trivial Bound’

L2 estimate n

d−1 2

  • |R|=2−n

αR hR

  • ∞ 2−n
  • |R|=2−n

|αR| holds with the L2 norm on the left hand side Cauchy-Schwarz inequality every point is contained in ≈ nd−1 dyadic rectangles of area 2−n

Bilyk Discrepancy Function and the Small Ball Inequality

slide-44
SLIDE 44

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

Bilyk Discrepancy Function and the Small Ball Inequality

slide-45
SLIDE 45

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

This conjecture is related to:

Bilyk Discrepancy Function and the Small Ball Inequality

slide-46
SLIDE 46

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

This conjecture is related to: Irregularities of Distribution

Bilyk Discrepancy Function and the Small Ball Inequality

slide-47
SLIDE 47

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

This conjecture is related to: Irregularities of Distribution Approximation Theory

Bilyk Discrepancy Function and the Small Ball Inequality

slide-48
SLIDE 48

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

This conjecture is related to: Irregularities of Distribution Approximation Theory Probability Theory

Bilyk Discrepancy Function and the Small Ball Inequality

slide-49
SLIDE 49

The Small Ball Conjecture and Discrepancy

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

Bilyk Discrepancy Function and the Small Ball Inequality

slide-50
SLIDE 50

The Small Ball Conjecture and Discrepancy

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

Conjecture 2 DN∞ (log N)

d 2 Bilyk Discrepancy Function and the Small Ball Inequality

slide-51
SLIDE 51

The Small Ball Conjecture and Discrepancy

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

Conjecture 2 DN∞ (log N)

d 2

Notice that, in both conjectures, one gains a square root over the L2 estimate.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-52
SLIDE 52

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

Bilyk Discrepancy Function and the Small Ball Inequality

slide-53
SLIDE 53

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

Random choice of sign shows this is sharp.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-54
SLIDE 54

The Conjecture

Small Ball Conjecture For dimensions d ≥ 2, we have 2−n

  • |R|=2−n

|αR| n

1 2 (d−2)

  • |R|=2−n

αRhR

  • ∞

Random choice of sign shows this is sharp. Talagrand has proved this in d = 2. Another proof by Temlyakov.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-55
SLIDE 55

Main result

Theorem ( DB, Michael Lacey & Armen Vagharshakyan) In dimensions d ≥ 3 there is a η(d) > 0, such that for all choices

  • f coefficients αR we have

n

d−1 2 −η(d)

  • |R|=2−n

αRhR

  • ∞ 2−n
  • |R|=2−n

|αR| (1) Previously known: d = 3 J´

  • zsef Beck (1989):

n−η ← (log n)−1/8 .

Bilyk Discrepancy Function and the Small Ball Inequality

slide-56
SLIDE 56

A New Result for the Discrepancy Function in higher dimensions

Theorem (DB, M.Lacey, A.Vagharshakyan) For d ≥ 3 there is a choice of 0 < η(d) < 1

2 for which the

following estimate holds for all collections PN ⊂ [0, 1]d: DN∞ (log N)

d−1 2 +η(d) . Bilyk Discrepancy Function and the Small Ball Inequality

slide-57
SLIDE 57

A New Result for the Discrepancy Function in higher dimensions

Theorem (DB, M.Lacey, A.Vagharshakyan) For d ≥ 3 there is a choice of 0 < η(d) < 1

2 for which the

following estimate holds for all collections PN ⊂ [0, 1]d: DN∞ (log N)

d−1 2 +η(d) .

η(d) ≈ 1 d2

Bilyk Discrepancy Function and the Small Ball Inequality

slide-58
SLIDE 58

A New Result for the Discrepancy Function in higher dimensions

Theorem (DB, M.Lacey, A.Vagharshakyan) For d ≥ 3 there is a choice of 0 < η(d) < 1

2 for which the

following estimate holds for all collections PN ⊂ [0, 1]d: DN∞ (log N)

d−1 2 +η(d) .

η(d) ≈ 1 d2 Previously known: d = 3 J´

  • zsef Beck (1989):

(log N)η ← (log log N)1/8 .

Bilyk Discrepancy Function and the Small Ball Inequality

slide-59
SLIDE 59

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

Theorem In dimension 2, we have 2−n

  • |R|=2−n

|αR|

  • |R|=2−n

αRhR

  • ∞

Bilyk Discrepancy Function and the Small Ball Inequality

slide-60
SLIDE 60

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

Theorem In dimension 2, we have 2−n

  • |R|=2−n

|αR|

  • |R|=2−n

αRhR

  • ∞

The method of proof is by duality.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-61
SLIDE 61

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

Theorem In dimension 2, we have 2−n

  • |R|=2−n

|αR|

  • |R|=2−n

αRhR

  • ∞

The method of proof is by duality. Use Riesz products.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-62
SLIDE 62

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

Theorem In dimension 2, we have 2−n

  • |R|=2−n

|αR|

  • |R|=2−n

αRhR

  • ∞

The method of proof is by duality. Use Riesz products. In Dimension 2, there is a ‘Product Rule’: If |R| = |R′|, then, hR · hR′ is either 1R ±hR∩R′ .

Bilyk Discrepancy Function and the Small Ball Inequality

slide-63
SLIDE 63

Product Rule: Illustration

Bilyk Discrepancy Function and the Small Ball Inequality

slide-64
SLIDE 64

Product Rule Fails in higher dimensions!!!

Bilyk Discrepancy Function and the Small Ball Inequality

slide-65
SLIDE 65

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

H def =

  • |R|=2−n

αR hR

Bilyk Discrepancy Function and the Small Ball Inequality

slide-66
SLIDE 66

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

H def =

  • |R|=2−n

αR hR f(k,n−k)

def

=

  • |R1|=2−k , |R2|=2−n+k

sgn(αR) hR , 0 ≤ k ≤ n ,

Bilyk Discrepancy Function and the Small Ball Inequality

slide-67
SLIDE 67

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

H def =

  • |R|=2−n

αR hR f(k,n−k)

def

=

  • |R1|=2−k , |R2|=2−n+k

sgn(αR) hR , 0 ≤ k ≤ n , F def =

n

  • k=0
  • 1 + f(k,n−k)
  • Bilyk

Discrepancy Function and the Small Ball Inequality

slide-68
SLIDE 68

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

H def =

  • |R|=2−n

αR hR f(k,n−k)

def

=

  • |R1|=2−k , |R2|=2−n+k

sgn(αR) hR , 0 ≤ k ≤ n , F def =

n

  • k=0
  • 1 + f(k,n−k)
  • F ≥ 0

&

  • F = 1

⇒ F1 = 1

Bilyk Discrepancy Function and the Small Ball Inequality

slide-69
SLIDE 69

Proof of Talagrand’s Theorem, following Temlyakov, Schmidt and Halasz

H def =

  • |R|=2−n

αR hR f(k,n−k)

def

=

  • |R1|=2−k , |R2|=2−n+k

sgn(αR) hR , 0 ≤ k ≤ n , F def =

n

  • k=0
  • 1 + f(k,n−k)
  • F ≥ 0

&

  • F = 1

⇒ F1 = 1 H∞ ≥ H, F =

n

  • k=0

H, f(k,n−k) = 2−n−1

  • |R|=2−n

|αR|

Bilyk Discrepancy Function and the Small Ball Inequality

slide-70
SLIDE 70

Halasz’s Proof of Schmidt’s Theorem

Theorem (Schmidt) For d = 2 we have DN∞ log N Take n ≈ log2 N and consider r = (k, n − k)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-71
SLIDE 71

Halasz’s Proof of Schmidt’s Theorem

Theorem (Schmidt) For d = 2 we have DN∞ log N Take n ≈ log2 N and consider r = (k, n − k) Use r functions such that DN, fr 1

Bilyk Discrepancy Function and the Small Ball Inequality

slide-72
SLIDE 72

Halasz’s Proof of Schmidt’s Theorem

Theorem (Schmidt) For d = 2 we have DN∞ log N Take n ≈ log2 N and consider r = (k, n − k) Use r functions such that DN, fr 1 F def = n

k=0

  • 1 + cf(k,n−k)
  • − 1

Bilyk Discrepancy Function and the Small Ball Inequality

slide-73
SLIDE 73

Halasz’s Proof of Schmidt’s Theorem

Theorem (Schmidt) For d = 2 we have DN∞ log N Take n ≈ log2 N and consider r = (k, n − k) Use r functions such that DN, fr 1 F def = n

k=0

  • 1 + cf(k,n−k)
  • − 1

F1 ≤ 2

Bilyk Discrepancy Function and the Small Ball Inequality

slide-74
SLIDE 74

Halasz’s Proof of Schmidt’s Theorem

Theorem (Schmidt) For d = 2 we have DN∞ log N Take n ≈ log2 N and consider r = (k, n − k) Use r functions such that DN, fr 1 F def = n

k=0

  • 1 + cf(k,n−k)
  • − 1

F1 ≤ 2 DN∞ DN, F n + higher order terms

Bilyk Discrepancy Function and the Small Ball Inequality

slide-75
SLIDE 75

The ‘Short’ Riesz Product

Set q = nǫ, this will be the length of our product.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-76
SLIDE 76

The ‘Short’ Riesz Product

Set q = nǫ, this will be the length of our product. Divide the integers {1, 2, . . . , n} into q disjoint intervals I1, . . . , Iq, and let It

def

= { r ∈ Nn : r1 ∈ It}.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-77
SLIDE 77

The ‘Short’ Riesz Product

Set q = nǫ, this will be the length of our product. Divide the integers {1, 2, . . . , n} into q disjoint intervals I1, . . . , Iq, and let It

def

= { r ∈ Nn : r1 ∈ It}. Ft =

  • r∈It

f

r .

ρ = q1/2 n(d−1)/2 true L2normalization.

  • ρ =

aqb n(d−1)/2 ’false’ normalization (b = 1/4, a << 1).

Bilyk Discrepancy Function and the Small Ball Inequality

slide-78
SLIDE 78

The ‘Short’ Riesz Product

Set q = nǫ, this will be the length of our product. Divide the integers {1, 2, . . . , n} into q disjoint intervals I1, . . . , Iq, and let It

def

= { r ∈ Nn : r1 ∈ It}. Ft =

  • r∈It

f

r .

ρ = q1/2 n(d−1)/2 true L2normalization.

  • ρ =

aqb n(d−1)/2 ’false’ normalization (b = 1/4, a << 1). Ψ def =

q

  • t=1

(1 + ρ Ft) .

Bilyk Discrepancy Function and the Small Ball Inequality

slide-79
SLIDE 79

Definition Vectors rj ∈ Nn, say that r1, . . . , rJ are strongly distinct iff there are no two vectors which are equal in any coordinate.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-80
SLIDE 80

Definition Vectors rj ∈ Nn, say that r1, . . . , rJ are strongly distinct iff there are no two vectors which are equal in any coordinate. Lemma The product of strongly distinct r functions is again an r function.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-81
SLIDE 81

Definition Vectors rj ∈ Nn, say that r1, . . . , rJ are strongly distinct iff there are no two vectors which are equal in any coordinate. Lemma The product of strongly distinct r functions is again an r function. Write Ψ = 1 + Ψsd + Ψ¬sd Ψsd = Sum of the strongly distinct products in the expansion of Ψ.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-82
SLIDE 82

Definition Vectors rj ∈ Nn, say that r1, . . . , rJ are strongly distinct iff there are no two vectors which are equal in any coordinate. Lemma The product of strongly distinct r functions is again an r function. Write Ψ = 1 + Ψsd + Ψ¬sd Ψsd = Sum of the strongly distinct products in the expansion of Ψ. We ‘only’ need to show that Ψsd1 1.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-83
SLIDE 83

Methods

Conditional expectation arguments

Bilyk Discrepancy Function and the Small Ball Inequality

slide-84
SLIDE 84

Methods

Conditional expectation arguments Lp estimates

Bilyk Discrepancy Function and the Small Ball Inequality

slide-85
SLIDE 85

Methods

Conditional expectation arguments Lp estimates Littlewood-Paley inequalities with sharp constants in p (Burkholder; Wang; R. Fefferman, J. Pipher)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-86
SLIDE 86

Methods

Conditional expectation arguments Lp estimates Littlewood-Paley inequalities with sharp constants in p (Burkholder; Wang; R. Fefferman, J. Pipher) Exponential moments (‘hyperbolic’ Chang-Wilson-Wolf, subgaussian estimates)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-87
SLIDE 87

Methods

Conditional expectation arguments Lp estimates Littlewood-Paley inequalities with sharp constants in p (Burkholder; Wang; R. Fefferman, J. Pipher) Exponential moments (‘hyperbolic’ Chang-Wilson-Wolf, subgaussian estimates) Analysis of ‘coincidences’

Bilyk Discrepancy Function and the Small Ball Inequality

slide-88
SLIDE 88

The Crucial Lemma of Beck

Lemma Beck Gain: We have the estimate

  • r=

s∈Nn r2=s2

f

r · f s

  • p p3/2n3/2

Bilyk Discrepancy Function and the Small Ball Inequality

slide-89
SLIDE 89

Restricted inequality in all dimensions

Theorem ( DB, Michael Lacey & Armen Vagharshakyan) In dimensions d ≥ 3 there is a η(d) > 0, such that for all choices

  • f coefficients αR ∈ {±1} we have
  • |R|=2−n

αRhR

  • ∞ n

d−1 2 +η(d)

(2) For d = 3, η = 1

10 − ε.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-90
SLIDE 90

Restricted inequality in all dimensions

Theorem ( DB, Michael Lacey & Armen Vagharshakyan) In dimensions d ≥ 3 there is a η(d) > 0, such that for all choices

  • f coefficients αR ∈ {±1} we have
  • |R|=2−n

αRhR

  • ∞ n

d−1 2 +η(d)

(2) For d = 3, η = 1

10 − ε.

In higher dimensions, η(d) ≈ 1

d .

Bilyk Discrepancy Function and the Small Ball Inequality

slide-91
SLIDE 91

Covering Numbers of Mixed Derivative Spaces

Let MD = Dx1 · · · Dxd be the mixed derivative operator.

Bilyk Discrepancy Function and the Small Ball Inequality

slide-92
SLIDE 92

Covering Numbers of Mixed Derivative Spaces

Let MD = Dx1 · · · Dxd be the mixed derivative operator. Define Balls

def

= {f : f s + MD f s ≤ 1}

Bilyk Discrepancy Function and the Small Ball Inequality

slide-93
SLIDE 93

Covering Numbers of Mixed Derivative Spaces

Let MD = Dx1 · · · Dxd be the mixed derivative operator. Define Balls

def

= {f : f s + MD f s ≤ 1} Define Ns(ǫ) := least number of L∞ balls of radius ǫ needed to cover Balls

Bilyk Discrepancy Function and the Small Ball Inequality

slide-94
SLIDE 94

Covering Numbers of Mixed Derivative Spaces

Let MD = Dx1 · · · Dxd be the mixed derivative operator. Define Balls

def

= {f : f s + MD f s ≤ 1} Define Ns(ǫ) := least number of L∞ balls of radius ǫ needed to cover Balls Conjecture For d ≥ 2, one has the estimate log N2(ǫ) ≈ 1

ǫ

  • log 1

ǫ

d−1/2

Bilyk Discrepancy Function and the Small Ball Inequality

slide-95
SLIDE 95

Covering Numbers of Mixed Derivative Spaces

Let MD = Dx1 · · · Dxd be the mixed derivative operator. Define Balls

def

= {f : f s + MD f s ≤ 1} Define Ns(ǫ) := least number of L∞ balls of radius ǫ needed to cover Balls Conjecture For d ≥ 2, one has the estimate log N2(ǫ) ≈ 1

ǫ

  • log 1

ǫ

d−1/2 Upper bound is known

Bilyk Discrepancy Function and the Small Ball Inequality

slide-96
SLIDE 96

Covering Numbers of Mixed Derivative Spaces

Let MD = Dx1 · · · Dxd be the mixed derivative operator. Define Balls

def

= {f : f s + MD f s ≤ 1} Define Ns(ǫ) := least number of L∞ balls of radius ǫ needed to cover Balls Conjecture For d ≥ 2, one has the estimate log N2(ǫ) ≈ 1

ǫ

  • log 1

ǫ

d−1/2 Upper bound is known Small Ball inequality for smooth wavelets implies the lower bound (Talagrand)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-97
SLIDE 97

Covering Numbers of Mixed Derivative Spaces

Let MD = Dx1 · · · Dxd be the mixed derivative operator. Define Balls

def

= {f : f s + MD f s ≤ 1} Define Ns(ǫ) := least number of L∞ balls of radius ǫ needed to cover Balls Conjecture For d ≥ 2, one has the estimate log N2(ǫ) ≈ 1

ǫ

  • log 1

ǫ

d−1/2 Upper bound is known Small Ball inequality for smooth wavelets implies the lower bound (Talagrand) Small Ball Inequality for Haars implies a bound for N1(ǫ)

Bilyk Discrepancy Function and the Small Ball Inequality

slide-98
SLIDE 98

Small Ball Problem for the Brownian Sheet

Let B : [0, 1]d − → R be the Brownian Sheet: EB(s)B(t) =

d

  • j=1

min{sj, tj}

Bilyk Discrepancy Function and the Small Ball Inequality

slide-99
SLIDE 99

Small Ball Problem for the Brownian Sheet

Let B : [0, 1]d − → R be the Brownian Sheet: EB(s)B(t) =

d

  • j=1

min{sj, tj} Theorem (Kuelbs, Li) − log P(BC[0,1]d < ǫ) ≈ ǫ−2 log 1

ǫ

β iff log N2(ǫ) ≈ ǫ−1 log 1

ǫ

β/2

Bilyk Discrepancy Function and the Small Ball Inequality

slide-100
SLIDE 100

Small Ball Problem for the Brownian Sheet

Let B : [0, 1]d − → R be the Brownian Sheet: EB(s)B(t) =

d

  • j=1

min{sj, tj} Theorem (Kuelbs, Li) − log P(BC[0,1]d < ǫ) ≈ ǫ−2 log 1

ǫ

β iff log N2(ǫ) ≈ ǫ−1 log 1

ǫ

β/2 Small Ball Problem For d ≥ 2, we have − log P(BC[0,1]d < ǫ) ≈ ǫ−2 log 1

ǫ

2d−1

Bilyk Discrepancy Function and the Small Ball Inequality