The small ball inequality and binary nets Naomi Feldheim (Stanford - PowerPoint PPT Presentation

The small ball inequality and binary nets Naomi Feldheim (Stanford University) j.w. Dmitriy Bilyk (University of Minnesota) IMU meeting Dead Sea, June 2016

Outline What is the “Small Ball Inequality” (SBI) Motivation Results: new connection with nets Proofs Ideas for higher dimensions Related methods in analysis

Definitions m = 0 , 1 , . . . , 2 k − 1 } D = { [ m 2 k , m +1 2 k ) : k ∈ N 0 , I ∈ D − → h I = − 1 I I left + 1 I I right . Note that � h I � ∞ = 1.

Definitions m = 0 , 1 , . . . , 2 k − 1 } D = { [ m 2 k , m +1 2 k ) : k ∈ N 0 , I ∈ D − → h I = − 1 I I left + 1 I I right . Note that � h I � ∞ = 1. D d = { R 1 × · · · × R d : R i ∈ D }

Definitions m = 0 , 1 , . . . , 2 k − 1 } D = { [ m 2 k , m +1 2 k ) : k ∈ N 0 , I ∈ D − → h I = − 1 I I left + 1 I I right . Note that � h I � ∞ = 1. D d = { R 1 × · · · × R d : R i ∈ D } R ∈ D d − → h R ( x 1 , . . . , x d ) = � d j =1 h R j ( x j )

Definitions m = 0 , 1 , . . . , 2 k − 1 } D = { [ m 2 k , m +1 2 k ) : k ∈ N 0 , I ∈ D − → h I = − 1 I I left + 1 I I right . Note that � h I � ∞ = 1. D d = { R 1 × · · · × R d : R i ∈ D } R ∈ D d − → h R ( x 1 , . . . , x d ) = � d j =1 h R j ( x j ) -1 1 1 -1

The small ball inequality Conjecture: Small Ball Inequality (SBI) Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � ∞ � 2 − n � n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n

The small ball inequality Conjecture: Small Ball Inequality (SBI) Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � ∞ � 2 − n � n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n The constant in � depends on d , not on n “reverse triangle inequality”

The small ball inequality Conjecture: Small Ball Inequality (SBI) Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � � ∞ � 2 − n n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n Conjecture: Signed Small Ball Inequality (SSBI) Let d ≥ 2. For any n ∈ N and ε R ∈ {± 1 } : � � d � ε R h R ∞ � n � � 2 � � | R | =2 − n

The small ball inequality Conjecture: Small Ball Inequality (SBI) Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � � ∞ � 2 − n n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n Conjecture: Signed Small Ball Inequality (SSBI) Let d ≥ 2. For any n ∈ N and ε R ∈ {± 1 } : � � d � ε R h R ∞ � n � � 2 � � | R | =2 − n SBI ⇒ SSBI: Notice: | ε R | = # { R ∈ D d : | R | = 2 − n } ≍ n d − 1 · 2 n � (=shape · placement).

An L 2 estimate Notice that � h R � 2 2 = | R | , and � h R 1 , h R 2 � = 0 for R 1 � = R 2 .

An L 2 estimate Notice that � h R � 2 2 = | R | , and � h R 1 , h R 2 � = 0 for R 1 � = R 2 . 1 / 2   � � �  � | α R | 2 2 − n α R h R 2 = � �  � � | R | =2 − n | R | =2 − n � | α R | 2 − n/ 2 C − S � ( n d − 1 2 n ) 1 / 2 = n − d − 1 2 2 − n � | α R | | R | =2 − n

An L 2 estimate Notice that � h R � 2 2 = | R | , and � h R 1 , h R 2 � = 0 for R 1 � = R 2 . 1 / 2   � � �  � | α R | 2 2 − n α R h R 2 = � �  � � | R | =2 − n | R | =2 − n � | α R | 2 − n/ 2 C − S � ( n d − 1 2 n ) 1 / 2 = n − d − 1 2 2 − n � | α R | | R | =2 − n SBI - L 2 estimate Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 1 � ∞ � 2 − n � n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n

An L 2 estimate SBI - L 2 estimate Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 1 � � ∞ � 2 − n n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n SSBI - L 2 estimate Let d ≥ 2. For any n ∈ N and ε R ∈ {± 1 } : � � d − 1 � ε R h R ∞ � n � � 2 � � | R | =2 − n

Struggle for power SBI - conjecture Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � � ∞ � 2 − n n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n SSBI - conjecture Let d ≥ 2. For any n ∈ N and ε R ∈ {± 1 } : � � d � ε R h R ∞ � n � � 2 � � | R | =2 − n

Struggle for power SBI - conjecture Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � � ∞ � 2 − n n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n SSBI - conjecture Let d ≥ 2. For any n ∈ N and ε R ∈ {± 1 } : � � d � ε R h R ∞ � n � � 2 � � | R | =2 − n d = 2: Talagrand ‘94; Temlyakov ‘95.

Struggle for power SBI - conjecture Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � � ∞ � 2 − n n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n SSBI - conjecture Let d ≥ 2. For any n ∈ N and ε R ∈ {± 1 } : � � d � ε R h R ∞ � n � � 2 � � | R | =2 − n d = 2: Talagrand ‘94; Temlyakov ‘95. Tightness: random ± 1 / Gaussians.

Struggle for power SBI - conjecture Let d ≥ 2. For any n ∈ N and α R ∈ R : � � d − 2 � � ∞ � 2 − n n α R h R | α R | � � 2 � � | R | =2 − n | R | =2 − n SSBI - conjecture Let d ≥ 2. For any n ∈ N and ε R ∈ {± 1 } : � � d � ε R h R ∞ � n � � 2 � � | R | =2 − n d = 2: Talagrand ‘94; Temlyakov ‘95. Tightness: random ± 1 / Gaussians. best power known: d − 1 + η ( d ) for d ≥ 3 2 (Bilyk-Lacey-Vagharshakyan 2008)

Motivation 1: Probability Let X t : T → R be a random process (usually Gaussian), estimate the small ball probability � � sup | X t | < ε ≈ ? , ε → 0 . P t ∈ T

Motivation 1: Probability Let X t : T → R be a random process (usually Gaussian), estimate the small ball probability � � sup | X t | < ε ≈ ? , ε → 0 . P t ∈ T The Brownian sheet in R d + is a Gaussian process B ( t ) with E ( B ( s ) B ( t )) = � d j =1 min( s j , t j ).

Motivation 1: Probability Let X t : T → R be a random process (usually Gaussian), estimate the small ball probability � � sup | X t | < ε ≈ ? , ε → 0 . P t ∈ T The Brownian sheet in R d + is a Gaussian process B ( t ) with E ( B ( s ) B ( t )) = � d j =1 min( s j , t j ). Conjecture (Talagrand): SB for Brownian sheet, d ≥ 2 � 2 d − 1 log 1 − log P (sup t ∈ [0 , 1] d | B ( t ) | < ε ) ≈ ε − 2 � ε

Motivation 1: Probability Let X t : T → R be a random process (usually Gaussian), estimate the small ball probability � � sup | X t | < ε ≈ ? , ε → 0 . P t ∈ T The Brownian sheet in R d + is a Gaussian process B ( t ) with E ( B ( s ) B ( t )) = � d j =1 min( s j , t j ). Conjecture (Talagrand): SB for Brownian sheet, d ≥ 2 � 2 d − 1 log 1 − log P (sup t ∈ [0 , 1] d | B ( t ) | < ε ) ≈ ε − 2 � ε � 2 d − 2 (Cs´ The L 2 estimate is ε − 2 � log 1 aki, ‘82) ε

Motivation 1: Probability Let X t : T → R be a random process (usually Gaussian), estimate the small ball probability � � sup | X t | < ε ≈ ? , ε → 0 . P t ∈ T The Brownian sheet in R d + is a Gaussian process B ( t ) with E ( B ( s ) B ( t )) = � d j =1 min( s j , t j ). Conjecture (Talagrand): SB for Brownian sheet, d ≥ 2 � 2 d − 1 log 1 − log P (sup t ∈ [0 , 1] d | B ( t ) | < ε ) ≈ ε − 2 � ε � 2 d − 2 (Cs´ The L 2 estimate is ε − 2 � log 1 aki, ‘82) ε known in d = 2 (Talagrand ‘94).

Motivation 1: Probability Let X t : T → R be a random process (usually Gaussian), estimate the small ball probability � � sup | X t | < ε ≈ ? , ε → 0 . P t ∈ T The Brownian sheet in R d + is a Gaussian process B ( t ) with E ( B ( s ) B ( t )) = � d j =1 min( s j , t j ). Conjecture (Talagrand): SB for Brownian sheet, d ≥ 2 � 2 d − 1 log 1 − log P (sup t ∈ [0 , 1] d | B ( t ) | < ε ) ≈ ε − 2 � ε � 2 d − 2 (Cs´ The L 2 estimate is ε − 2 � log 1 aki, ‘82) ε known in d = 2 (Talagrand ‘94). � 2 d − 2+ η , with some η ( d ) > 0 log 1 LB: ε − 2 � ε (Bilyk-Lacey-Vagharshakyan ‘08). Method: write B ( t ) in “wavelet” basis and use modified SBI

Motivation 1: Probability Let X t : T → R be a random process (usually Gaussian), estimate the small ball probability � � sup | X t | < ε ≈ ? , ε → 0 . P t ∈ T The Brownian sheet in R d + is a Gaussian process B ( t ) with E ( B ( s ) B ( t )) = � d j =1 min( s j , t j ). Conjecture (Talagrand): SB for Brownian sheet, d ≥ 2 � 2 d − 1 log 1 − log P (sup t ∈ [0 , 1] d | B ( t ) | < ε ) ≈ ε − 2 � ε � 2 d − 2 (Cs´ The L 2 estimate is ε − 2 � log 1 aki, ‘82) ε known in d = 2 (Talagrand ‘94). � 2 d − 2+ η , with some η ( d ) > 0 log 1 LB: ε − 2 � ε (Bilyk-Lacey-Vagharshakyan ‘08). Method: write B ( t ) in “wavelet” basis and use modified SBI SBP ↔ metric entropy (Kuelbs-Li ‘93)

Motivation 2: Discrepancy Theory How well can a set of N points be “equidistributed” in the d -dimensional cube?

Motivation 2: Discrepancy Theory How well can a set of N points be “equidistributed” in the d -dimensional cube? Consider a set P N ⊂ [0 , 1] d consisting of N points: