On some sets with minimal L 2 discrepancy Dmitriy Bilyk University - PowerPoint PPT Presentation

On some sets with minimal L 2 discrepancy Dmitriy Bilyk University of South Carolina, Columbia, SC MCQMC 2010 Warszawa, Polska August 20, 2010 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Discrepancy function Consider P N ⊂ [0 , 1] d with # P N = N : D N ( x ) = ♯ {P N ∩ [0 , x ) } − Nx 1 x 2 . . . x d Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L ∞ estimates (star-discrepancy) Theorem (Schmidt, 1972) For d = 2 we have � D N � ∞ � log N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L ∞ estimates (star-discrepancy) Theorem (Schmidt, 1972) For d = 2 we have � D N � ∞ � log N d = 2, Lerch (1904); van der Corput: There exist P N ⊂ [0 , 1] 2 with � D N � ∞ ≈ log N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L ∞ estimates (star-discrepancy) Theorem (Schmidt, 1972) For d = 2 we have � D N � ∞ � log N d = 2, Lerch (1904); van der Corput: There exist P N ⊂ [0 , 1] 2 with � D N � ∞ ≈ log N d ≥ 3, Halton (1960); Hammersely; Niederreiter; Faure There exist P N ⊂ [0 , 1] d with � D N � ∞ � (log N ) d − 1 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L ∞ estimates (star-discrepancy) Theorem (Schmidt, 1972) For d = 2 we have � D N � ∞ � log N d = 2, Lerch (1904); van der Corput: There exist P N ⊂ [0 , 1] 2 with � D N � ∞ ≈ log N d ≥ 3, Halton (1960); Hammersely; Niederreiter; Faure There exist P N ⊂ [0 , 1] d with � D N � ∞ � (log N ) d − 1 Theorem (DB, Lacey, Vagharshakyan, 2007) For d ≥ 3 there exists 0 < ε d ≤ 1 2 , such that d − 1 2 + ε d � D N � ∞ � (log N ) Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L p estimates, 1 < p < ∞ Theorem d − 1 � D N � p � (log N ) 2 Roth (p = 2 ) 1954, Schmidt (p � = 2 ) 1977 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L p estimates, 1 < p < ∞ Theorem d − 1 � D N � p � (log N ) 2 Roth (p = 2 ) 1954, Schmidt (p � = 2 ) 1977 Theorem There exist P N ⊂ [0 , 1] d with d − 1 � D N � p ≈ (log N ) 2 (Davenport; Roth; Halton, Zaremba; Chen, Skriganov) Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Van der Corput set “Digit reversing” van der Corput set Denote the binary expansion of x ∈ [0 , 1) by � x i · 2 − i = 0 . x 1 x 2 ... x n ... x = i The van der Corput set V n with 2 n points is defined as: V n = { ( 0 . x 1 x 2 . . . x n − 1 x n , 0 . x n x n − 1 . . . x 2 x 1 ) : x i = 0 , 1 } Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 3 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 4 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 5 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 6 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 7 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 8 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 9 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 10 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 11 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

van der Corput set van der Corput set with N = 2 12 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Van der Corput set “Digit reversing” van der Corput set Denote the binary expansion of x ∈ [0 , 1) by � x i · 2 − i = 0 . x 1 x 2 ... x n ... x = i The van der Corput set V n with 2 n points is defined as: V n = { ( 0 . x 1 x 2 . . . x n − 1 x n , 0 . x n x n − 1 . . . x 2 x 1 ) : x i = 0 , 1 } Theorem (van der Corput) The set V n satisfies with � D V n � ∞ � n ≈ log N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Irrational lattice Example Let α be an irrational number and let { x } denote the fractional part of x . Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Irrational lattice Example Let α be an irrational number and let { x } denote the fractional �� i �� N − 1 part of x . Define P N = N , { i α } i =0 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Irrational lattice Example Let α be an irrational number and let { x } denote the fractional �� i �� N − 1 part of x . Define P N = N , { i α } i =0 Theorem If the partial quotients of the continued fraction of α are bounded, then the discrepancy function of this set satisfies � D N � ∞ ≈ log N . Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Irrational lattice Example Let α be an irrational number and let { x } denote the fractional �� i �� N − 1 part of x . Define P N = N , { i α } i =0 Theorem If the partial quotients of the continued fraction of α are bounded, then the discrepancy function of this set satisfies � D N � ∞ ≈ log N . In particular works for quadratic irrationalities α = u + √ v . Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Irrational lattice Example Let α be an irrational number and let { x } denote the fractional �� i �� N − 1 part of x . Define P N = N , { i α } i =0 Theorem If the partial quotients of the continued fraction of α are bounded, then the discrepancy function of this set satisfies � D N � ∞ ≈ log N . In particular works for quadratic irrationalities α = u + √ v . The idea goes as far back as 1904 (Lerch) Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Low discrepancy sets √ 2) lattice with N = 2 12 points The irrational ( α = Discrepancy ≈ log N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Low discrepancy sets The van der Corput set with N = 2 12 points Discrepancy ≈ log N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Low discrepancy sets Random set with N = 2 12 points √ Discrepancy ≈ N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L 2 bounds Theorem (K. Roth) In dimension d = 2 , for any N-point set P N ⊂ [0 , 1] 2 , � � � � D N � 2 � log N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

L 2 bounds Theorem (K. Roth) In dimension d = 2 , for any N-point set P N ⊂ [0 , 1] 2 , � � � � D N � 2 � log N Standard sets fail to meet this bound For the van der Corput set and the irrational lattice, we have � � � D N � 2 ≈ log N Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedies 1. Davenport’s reflection (symmetrization) Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedies 1. Davenport’s reflection (symmetrization) 2. Digital shifts (digit-scrambling) Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedies 1. Davenport’s reflection (symmetrization) 2. Digital shifts (digit-scrambling) 3. Cyclic shifts (mod 1) Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedies 1. Davenport’s reflection (symmetrization) 2. Digital shifts (digit-scrambling) 3. Cyclic shifts (mod 1) Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedy: Cyclic shifts �� � � Define V α n = ( x + α ) mod 1 , y : ( x , y ) ∈ V n van der Corput set with N = 2 8 points Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedy: Cyclic shifts �� � � Define V α n = ( x + α ) mod 1 , y : ( x , y ) ∈ V n van der Corput set with N = 2 8 points translated (mod 1) by 1 / 8 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedy: Cyclic shifts �� � � Define V α n = ( x + α ) mod 1 , y : ( x , y ) ∈ V n van der Corput set with N = 2 8 points translated (mod 1) by 2 / 8 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedy: Cyclic shifts �� � � Define V α n = ( x + α ) mod 1 , y : ( x , y ) ∈ V n van der Corput set with N = 2 8 points translated (mod 1) by 3 / 8 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedy: Cyclic shifts �� � � Define V α n = ( x + α ) mod 1 , y : ( x , y ) ∈ V n van der Corput set with N = 2 8 points translated (mod 1) by 4 / 8 Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedy: Cyclic shifts �� � � Define V α n = ( x + α ) mod 1 , y : ( x , y ) ∈ V n Theorem (K. Roth, 1979) � � � � D V α � log N E α 2 � n Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis

Remedy: Cyclic shifts �� � � Define V α n = ( x + α ) mod 1 , y : ( x , y ) ∈ V n Theorem (K. Roth, 1979) � � � � D V α � log N E α 2 � n Theorem (D.B., 2008) For α = 1 − k 2 n , where � � n 2 = 54 n 1 k = 000111 ... 000111 � 00001111 ... 00001111 � �� � �� � 17 2 n 1 digits n 2 digits we have � � � � D V α � 2 � log N n Dmitriy Bilyk Geometric Discrepancy and Harmonic Analysis