Lower Bounds for L 1 Discrepancy Armen Vagharshakyan Brown - - PowerPoint PPT Presentation

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Lower Bounds for L1 Discrepancy

Armen Vagharshakyan

Brown University

January 10, 2013

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Discrepancy Function

P ⊂ [0, 1]2 - a finite set. DP(x, y) = ♯(P ∩ [0, x] × [0, y]) − ♯(P) · xy, (x, y) ∈ [0, 1]2. The discrepancy function measures the difference between the actual number of points of the set P in an axis-parallel rectangle [0; x] × [0; y] and the “expected” number of points in that

• rectangle. Thus, the discrepancy function quantifies the

“closeness” of the distribution generated by a finite set P to the uniform distribution.

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Koksma-Hlawka Inequality

• 1

1 f (x, y) dxdy − 1 N

• x∈P

f (x)

• ≤ V (f ) · ||DP||∞

N Other applications of discrepancy: Uniformly distributed sequences, Metric entropy, Brownian process. . .

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Estimates for L1 discrepancy

dN = 1 √ ln N · inf

♯(P)=N

1 1 |DP(x, y)| dxdy

• H. Davenport (1956): lim supN→∞ dN < +∞
• G. Halasz (1981) [the only lower estimate known before]:

lim inf

N→∞ dN ≥ 0.00039 > 0

A.V. (2012) lim inf

N→∞ dN ≥

3 256 √ e ln 2 ≈ 0.00854 lim sup

N→∞

dN ≥ 1 64 √ e ln 2 ≈ 0.01137

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Lower bounds for L1 and L2 discrepancy

A.V. (2012) lim inf

N→∞

1 √ ln N · inf

♯(P)=N ||DP||L1([0,1]2) ≥

3 256 √ e ln 2 ≈ 0.00854 lim sup

N→∞

1 √ ln N · inf

♯(P)=N ||DP||L1([0,1]2) ≥

1 64 √ e ln 2 ≈ 0.01137 Hinrichs, Markashin (2011) lim inf

N→∞

1 √ ln N · inf

♯(P)=N ||DP||L2([0,1]2) ≥ 0.03276

lim sup

N→∞

1 √ ln N · inf

♯(P)=N ||DP||L2([0,1]2) ≥ 0.38930

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Optimal L2 discrepancy

Faure, Pillichshammer, Pirsic, Schmid (2010): lim inf

N→∞

1 √ ln N · inf

♯(P)=N ||DP||L2([0,1]2) ≤ 0.17905

Bilyk, Temlyakov, Yu (2012): lim inf

N→∞

1 √ ln N · inf

♯(P)=N ||DP||L2([0,1]2) ≤ 0.17601

For sets with optimal L2 discrepancy: |DP(x, y)| ≈ ||DP||L2([0,1]2)

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Properties of Auxiliary Functions

Corresponding to the set P, K.Roth (1954) constructed functions fi so that:

• 1. fi : [0, 1]2 → {−1, 0, 1}

2. 1 1 DP · fi < −c < 0

• 3. Orthogonality

Number of fi’s is around ♯(P)

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Roth orthogonal function method (as modified by Halasz)

||DP||1 ≥ 1 1

0 DP · Hn(f0, f1, . . . , fn)

||Hn(f0, f1, . . . , fn)||∞ ≥ LIN(Hn)

• n
• i=0

1 1 DPfi

• +Err

Hn is an odd, symmetric function, entire in each of its variables. Hn(f0, . . . , fn) = ∂Hn ∂x0

n

• i=0

fi + . . . LIN(Hn) = ∂Hn ∂x0

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Auxiliary Functions Revised

• 1. fi : [0, 1]2 → {−1, 1}

2. 1 1 Dpfi < −c < 0

• 3. Orthogonality

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Combinatorics of Auxiliary Functions

Hn(f0, f1, . . . , fn) = T f0 + f1 + · · · + fn √n

• =

= T ′(0) √n

n

• i=0

fi + T (3)(0) 3!(√n)3 n

• i=0

fi 3 + . . . (f0 + f1)3 = f 3

0 + 3f0f 2 1 + 3f 2 0 f1 + f 3 1 = 4f0 + 4f1

∗ combinatorial identities help us discuss a large family of test functions

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Main Inequality

• 1. T is odd
• 2. ||T||∞ = 1
• 3. the support of ˜

T is countable: {ωj}

• 4. {ωj} is linearly independent over Z

then lim sup √n · LIN

• T

f0 + · · · + fn √n

• =
• ∂e∂2/2

(T)(0) ≤ 1 √e and the maximum is obtained for the function: T(x) = sin(x)

Armen Vagharshakyan Lower Bounds for L1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks

Higher dimensions

Hn(f0, f1, . . . , fn) = T f0 + f1 + · · · + fn √n

• In higher dimensions we still have orthogonality of the functions fi,

but we don’t have independence anymore :(

Armen Vagharshakyan Lower Bounds for L1 Discrepancy