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

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

Geometric Discrepancy Let P N 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

Geometric Discrepancy Let P N 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 ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Bilyk Discrepancy Function and the Small Ball Inequality

Geometric Discrepancy Let P N 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 ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Star-discrepancy: D ( P N ) = sup | D ( P N , R ) | R Bilyk Discrepancy Function and the Small Ball Inequality

Geometric Discrepancy Let P N 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 ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Star-discrepancy: D ( P N ) = sup | D ( P N , R ) | R D ( N ) = inf P N D ( P N ) Bilyk Discrepancy Function and the Small Ball Inequality

Geometric Discrepancy Let P N 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 ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Star-discrepancy: D ( P N ) = sup | D ( P N , R ) | R D ( N ) = inf P N D ( P N ) Can D ( N ) be bounded? Bilyk Discrepancy Function and the Small Ball Inequality

Geometric Discrepancy Let P N 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 ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Star-discrepancy: D ( P N ) = sup | D ( P N , R ) | R D ( N ) = inf P N D ( P N ) Can D ( N ) be bounded? NO (van Aardenne-Ehrenfest; Roth) Bilyk Discrepancy Function and the Small Ball Inequality

Discrepancy function Enough to consider rectangles with a vertex at the origin Bilyk Discrepancy Function and the Small Ball Inequality

Discrepancy function Enough to consider rectangles with a vertex at the origin D N ( x ) = ♯ {P N ∩ [0 , x ) } − Nx 1 x 2 . . . x d Bilyk Discrepancy Function and the Small Ball Inequality

Discrepancy function Enough to consider rectangles with a vertex at the origin D N ( x ) = ♯ {P N ∩ [0 , x ) } − Nx 1 x 2 . . . x d Lower estimates for � D N � p ??? Bilyk Discrepancy Function and the Small Ball Inequality

Koksma-Hlawka Inequality � � � � � � � � [0 , 1] d f ( x ) dx − 1 � 1 � � f ( p ) N V ( f ) · � D N � ∞ � � N � � p ∈P N Bilyk Discrepancy Function and the Small Ball Inequality

Koksma-Hlawka Inequality � � � � � � � � [0 , 1] d f ( x ) dx − 1 � 1 � � f ( p ) N V ( f ) · � D N � ∞ � � N � � p ∈P N V ( f ) is the Hardy-Krause variation of f Bilyk Discrepancy Function and the Small Ball Inequality

Koksma-Hlawka Inequality � � � � � � � � [0 , 1] d f ( x ) dx − 1 � 1 � � f ( p ) N V ( f ) · � D N � ∞ � � N � � p ∈P N V ( f ) is the Hardy-Krause variation of f � � � � � ∂ d f V ( f ) ” = ” � dx 1 . . . dx d � [0 , 1] d ∂ x 1 ∂ x 2 ...∂ x d Bilyk Discrepancy Function and the Small Ball Inequality

An example √ √ Consider a N × N lattice Bilyk Discrepancy Function and the Small Ball Inequality

An example √ √ Consider a N × N lattice Bilyk Discrepancy Function and the Small Ball Inequality

An example √ √ Consider a N × N lattice √ 1 | D N ( P ) − D N ( Q ) | ≈ N · N = N √ Bilyk Discrepancy Function and the Small Ball Inequality

An example √ √ Consider a N × N lattice √ 1 | D N ( P ) − D N ( Q ) | ≈ N · N = N √ 1 Thus � D N � ∞ � N 2 Bilyk Discrepancy Function and the Small Ball Inequality

L p estimates, 1 < p < ∞ Theorem d − 1 � D N � p � (log N ) 2 Roth (p=2), Schmidt Bilyk Discrepancy Function and the Small Ball Inequality

L p estimates, 1 < p < ∞ Theorem d − 1 � D N � p � (log N ) 2 Roth (p=2), Schmidt Theorem There exist P N ⊂ [0 , 1] d with d − 1 � D N � p ≈ (log N ) 2 (Davenport, Roth, Frolov, Chen) Bilyk Discrepancy Function and the Small Ball Inequality

L ∞ estimates Conjecture d − 1 � D N � ∞ ≫ (log N ) 2 Bilyk Discrepancy Function and the Small Ball Inequality

L ∞ estimates Conjecture d − 1 � D N � ∞ ≫ (log N ) 2 Theorem (Schmidt) For d = 2 we have � D N � ∞ � log N Bilyk Discrepancy Function and the Small Ball Inequality

L ∞ estimates Conjecture d − 1 � D N � ∞ ≫ (log N ) 2 Theorem (Schmidt) For d = 2 we have � D N � ∞ � log N d = 2 van der Corput: There exist P N ⊂ [0 , 1] 2 with � D N � ∞ ≈ log N Bilyk Discrepancy Function and the Small Ball Inequality

L ∞ estimates Conjecture d − 1 � D N � ∞ ≫ (log N ) 2 Theorem (Schmidt) For d = 2 we have � D N � ∞ � log N d = 2 van der Corput: There exist P N ⊂ [0 , 1] 2 with � D N � ∞ ≈ log N d ≥ 3 Halasz: There exist P N ⊂ [0 , 1] d with � D N � ∞ ≈ (log N ) d − 1 Bilyk Discrepancy Function and the Small Ball Inequality

Conjectures Conjecture 1 � D N � ∞ � (log N ) d − 1 Bilyk Discrepancy Function and the Small Ball Inequality

Conjectures Conjecture 1 � D N � ∞ � (log N ) d − 1 Conjecture 2 d � D N � ∞ � (log N ) 2 Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Definitions Dyadic intervals are intervals of the form [ k 2 q , ( k + 1)2 q ). Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Definitions Dyadic intervals are intervals of the form [ k 2 q , ( k + 1)2 q ). For a dyadic interval I : h I = − 1 I left + 1 I right , Haar functions with L ∞ normalization. Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Definitions Dyadic intervals are intervals of the form [ k 2 q , ( k + 1)2 q ). For a dyadic interval I : h I = − 1 I left + 1 I right , Haar functions with L ∞ normalization. In higher dimensions: for a rectangle R = I 1 × · · · × I d h R ( x 1 , . . . , x d ) := h I 1 ( x 1 ) · ... · h I d ( x d ) Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Definitions All rectangles are in the unit cube in R d . Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Definitions All rectangles are in the unit cube in R d . Choose n appropriately so that n ≈ log 2 N and consider dyadic rectangles of volume 2 − n . Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Definitions All rectangles are in the unit cube in R d . Choose n appropriately so that n ≈ log 2 N and consider dyadic rectangles of volume 2 − n . n = { ( r 1 , r 2 , . . . , r d ) ∈ N d : r 1 + · · · + r d = n } H d Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Definitions All rectangles are in the unit cube in R d . Choose n appropriately so that n ≈ log 2 N and consider dyadic rectangles of volume 2 − n . n = { ( r 1 , r 2 , . . . , r d ) ∈ N d : r 1 + · · · + r d = n } H d Definition r ∈ H d For � n , call f an � r function iff it is of the form � f = ε R h R , ε R ∈ {± 1 } . R : | R j | =2 − rj Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Proof r ∈ H d For each � n , there exists f r such that � D N , f r � � 1. Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Proof r ∈ H d For each � n , there exists f r such that � D N , f r � � 1. Construct the test function F = � n f r . r ∈ H d � Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Proof r ∈ H d For each � n , there exists f r such that � D N , f r � � 1. Construct the test function F = � n f r . r ∈ H d � n } ≈ n d − 1 ≈ (log N ) d − 1 . r ∈ H d ♯ { � Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Proof r ∈ H d For each � n , there exists f r such that � D N , f r � � 1. Construct the test function F = � n f r . r ∈ H d � n } ≈ n d − 1 ≈ (log N ) d − 1 . r ∈ H d ♯ { � d − 1 2 . � F � 2 ≈ (log N ) Bilyk Discrepancy Function and the Small Ball Inequality

Roth’s Orthogonal Function Method: Proof r ∈ H d For each � n , there exists f r such that � D N , f r � � 1. Construct the test function F = � n f r . r ∈ H d � n } ≈ n d − 1 ≈ (log N ) d − 1 . r ∈ H d ♯ { � d − 1 2 . � F � 2 ≈ (log N ) � D N , F � � (log N ) d − 1 . Bilyk Discrepancy Function and the Small Ball Inequality