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 L 1 Discrepancy Armen Vagharshakyan Brown University January 10, 2013 Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Discrepancy Function P ⊂ [0 , 1] 2 - a finite set. ( x , y ) ∈ [0 , 1] 2 . D P ( x , y ) = ♯ ( P ∩ [0 , x ] × [0 , y ]) − ♯ ( P ) · xy , 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 L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Koksma-Hlawka Inequality � 1 � 1 � � f ( x , y ) dxdy − 1 � ≤ V ( f ) · || D P || ∞ � � � f ( x ) � � � N � N 0 0 � x ∈ P Other applications of discrepancy: Uniformly distributed sequences, Metric entropy, Brownian process. . . Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Estimates for L 1 discrepancy � 1 � 1 1 d N = √ · inf | D P ( x , y ) | dxdy ln N ♯ ( P )= N 0 0 H. Davenport (1956): lim sup N →∞ d N < + ∞ G. Halasz (1981) [the only lower estimate known before]: lim inf N →∞ d N ≥ 0 . 00039 > 0 A.V. (2012) 3 √ lim inf N →∞ d N ≥ ≈ 0 . 00854 256 e ln 2 1 lim sup d N ≥ √ ≈ 0 . 01137 64 e ln 2 N →∞ Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Lower bounds for L 1 and L 2 discrepancy A.V. (2012) 1 3 √ √ lim inf · ♯ ( P )= N || D P || L 1 ([0 , 1] 2 ) ≥ inf ≈ 0 . 00854 ln N 256 e ln 2 N →∞ 1 1 lim sup √ · ♯ ( P )= N || D P || L 1 ([0 , 1] 2 ) ≥ inf √ ≈ 0 . 01137 ln N 64 e ln 2 N →∞ Hinrichs, Markashin (2011) 1 √ lim inf · ♯ ( P )= N || D P || L 2 ([0 , 1] 2 ) ≥ 0 . 03276 inf ln N N →∞ 1 lim sup √ · ♯ ( P )= N || D P || L 2 ([0 , 1] 2 ) ≥ 0 . 38930 inf ln N N →∞ Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Optimal L 2 discrepancy Faure, Pillichshammer, Pirsic, Schmid (2010): 1 √ lim inf · ♯ ( P )= N || D P || L 2 ([0 , 1] 2 ) ≤ 0 . 17905 inf ln N N →∞ Bilyk, Temlyakov, Yu (2012): 1 √ lim inf · ♯ ( P )= N || D P || L 2 ([0 , 1] 2 ) ≤ 0 . 17601 inf ln N N →∞ For sets with optimal L 2 discrepancy: | D P ( x , y ) | ≈ || D P || L 2 ([0 , 1] 2 ) Armen Vagharshakyan Lower Bounds for L 1 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 f i so that: 1 . f i : [0 , 1] 2 → {− 1 , 0 , 1 } � 1 � 1 2 . D P · f i < − c < 0 0 0 3. Orthogonality Number of f i ’s is around ♯ ( P ) Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Roth orthogonal function method (as modified by Halasz) � 1 � 1 � 1 � 1 � � n 0 D P · H n ( f 0 , f 1 , . . . , f n ) � � � 0 || D P || 1 ≥ ≥ LIN ( H n ) D P f i � + Err � � || H n ( f 0 , f 1 , . . . , f n ) || ∞ � � 0 0 � i =0 H n is an odd, symmetric function, entire in each of its variables. n H n ( f 0 , . . . , f n ) = ∂ H n � f i + . . . ∂ x 0 i =0 LIN ( H n ) = ∂ H n ∂ x 0 Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Auxiliary Functions Revised 1 . f i : [0 , 1] 2 → {− 1 , 1 } � 1 � 1 2 . D p f i < − c < 0 0 0 3. Orthogonality Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Combinatorics of Auxiliary Functions � f 0 + f 1 + · · · + f n � √ n H n ( f 0 , f 1 , . . . , f n ) = T = � n � 3 n f i + T (3) (0) = T ′ (0) � � √ n 3!( √ n ) 3 f i + . . . i =0 i =0 ( f 0 + f 1 ) 3 = f 3 0 + 3 f 0 f 2 1 + 3 f 2 0 f 1 + f 3 1 = 4 f 0 + 4 f 1 ∗ combinatorial identities help us discuss a large family of test functions Armen Vagharshakyan Lower Bounds for L 1 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: { ω j } 4. { ω j } is linearly independent over Z then lim sup √ n · LIN � � f 0 + · · · + f n �� 1 � ∂ e ∂ 2 / 2 � √ n = ( T )(0) ≤ √ e T and the maximum is obtained for the function: T ( x ) = sin( x ) Armen Vagharshakyan Lower Bounds for L 1 Discrepancy

Introduction to discrepancy Estimates Roth’s method Elements of proof Closing remarks Higher dimensions � f 0 + f 1 + · · · + f n � √ n H n ( f 0 , f 1 , . . . , f n ) = T In higher dimensions we still have orthogonality of the functions f i , but we don’t have independence anymore :( Armen Vagharshakyan Lower Bounds for L 1 Discrepancy