L 2 discrepancy of digit scrambled two-dimensional Hammersley point - PowerPoint PPT Presentation

L 2 discrepancy of digit scrambled two-dimensional Hammersley point sets Friedrich Pillichshammer 1 Linz/Austria Joint work with Henri Faure (Marseille) Gottlieb Pirsic (Linz) Wolfgang Schmid (Salzburg) 1 Supported by the Austrian Science Foundation (FWF), Project S9609. Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 1 / 1

Discrepancy Let P = { x 0 , . . . , x N − 1 } ⊆ [0 , 1) 2 . For t ∈ [0 , 1] 2 set ∆ P ( t ) = # { 0 ≤ n < N : x n ∈ [ 0 , t ) } − N λ ([ 0 , t )) . Definition (discrepancy) For P ⊆ [0 , 1) 2 the L 2 -discrepancy is defined as � 1 / 2 �� [0 , 1] 2 | ∆ P ( t ) | 2 d t L 2 ( P ) := . Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 2 / 1

Bounds on L 2 Lower bound on L 2 (Roth 1954) ∃ c > 0 such that for any P ⊆ [0 , 1) 2 with # P = N we have � L 2 ( P ) ≥ c log N . E.g., c = 7 / (216 √ log 2) = 0 . 038925 . . . (Hinrichs, Markhasin, 2011). Existence result (Bilyk, Chaix, Chen, Davenport, Faure, Halton, Kritzer, Larcher, P., Pirsic, Proinov, Skriganov, Temlyakov, Roth, Schmid, White, Yu, Zaremba, ...) ∃ C > 0 such that for any N ∈ N , N ≥ 2, there exists P ⊆ [0 , 1) 2 with # P = N and � L 2 ( P ) ≤ C log N . Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 3 / 1

2-dimensional Hammersley point sets Let b ∈ N , b ≥ 2. For n ∈ N 0 with n = a 0 + a 1 b + a 2 b 2 + · · · define φ b ( n ) := a 0 b + a 1 b 2 + a 2 b 3 + · · · . Hammersley point set The 2-dimensional Hammersley point set is defined as φ b ( n ) , n �� � : 0 ≤ n < b m � H b , m := where m ∈ N 0 . b m Note: # H b , m = N = b m . Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 4 / 1

Hammersley point sets: examples Figure: Hammersley point sets with b = 2, m = 5 (left) and b = 3, m = 4 (right). Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 5 / 1

L 2 discrepancy of H b , m Theorem (Faure, P. 2009) For any b ≥ 2 and any m ∈ N 0 L 2 ( H b , m ) 2 � b 2 − 1 � 2 � 3 b 4 + 10 b 2 − 13 + b 2 − 1 � 1 �� m 2 = + m 1 − 720 b 2 2 b m 12 b 12 b + 3 1 1 8 + 4 b m − 72 b 2 m . In particular L 2 ( H b , m ) ≍ log N . Not best possible L 2 compared with Roth. Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 6 / 1

generalized Hammersley point set S b := S ( { 0 , 1 , . . . , b − 1 } ) (symmetric group); for m ∈ N 0 let Σ = ( σ 1 , . . . , σ m ) ∈ S m b ; for 0 ≤ n < b m with n = a 0 + a 1 b + · · · + a m − 1 b m − 1 define b ( n ) := σ 1 ( a 0 ) + σ 2 ( a 1 ) + · · · + σ m ( a m − 1 ) φ Σ . b 2 b m b generalized Hammersley point set Let m ∈ N 0 and let Σ ∈ S m b . The generalized 2-dimensional Hammersley point set is defined as b ( n ) , n �� � : 0 ≤ n < b m � H Σ φ Σ b , m := . b m If σ i ≡ id we obtain H b , m . Obviously: # H Σ b , m = N = b m Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 7 / 1

Some definitions For σ ∈ S b and h ∈ { 1 , 2 , . . . , b − 1 } we define ϕ σ b , h : [0 , 1) → R . � k − 1 b , k � If x ∈ , where k ∈ { 1 , . . . , b } , put b  # { 0 ≤ j < k : σ ( j ) < h } − hx if h ≤ σ ( k − 1) ,  ϕ σ b , h ( x ) := ( b − h ) x − # { 0 ≤ j < k : σ ( j ) ≥ h } if σ ( k − 1) < h .  Then � α � β m b m , β � � ϕ σ j � ∆ H Σ = where h j = h j ( α, β, m ) . b , h j b m b j b , m j =1 For r ∈ { 1 , 2 } put � 1 b − 1 := 1 ϕ σ, ( r ) � r and I σ, ( r ) ϕ σ, ( r ) � ϕ σ � := ( x ) d x . b , h b b b b 0 h =1 Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 8 / 1

Some definitions Example: b = 2, σ = (0 , 1) ∈ S 2 and h = 1: 0 , 1 � � x ∈ , i.e., k = 1 and σ ( k − 1) = σ (0) = 1. Hence 2 ϕ σ 2 , 1 ( x ) = # { 0 ≤ j < 1 : σ ( j ) < 1 } − x = − x . � 1 � x ∈ 2 , 1 , i.e., k = 2 and σ ( k − 1) = σ (1) = 0. Hence ϕ σ 2 , 1 ( x ) = x − # { 0 ≤ j < 2 : σ ( j ) ≥ 1 } = x − 1 . Hence ϕ σ, (1) ( x ) = ϕ σ 2 , 1 ( x ) = − min( x , 1 − x ) =: −� x � 2 and = − 1 I σ, (1) 8 . 2 Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 9 / 1

Some definitions For π ∈ S b define π ℓ ( k ) := π ( k ) + ℓ (mod b ) (linear scrambling). We consider Σ ∈ { π ℓ : 0 ≤ ℓ < b } m . White (1975): Σ = ( id 0 , id 1 , . . . , id b − 1 , id 0 , id 1 , . . . , id b − 1 , . . . ). Let τ b ∈ S b , τ b ( k ) := b − 1 − k (swapping permutation). For π ∈ S b we consider Σ ∈ { π, τ b ◦ π } m . Note that for b = 2, π = id we have id 1 = τ 2 (Halton & Zaremba (1969), Kritzer & P. (2006, 2007)). Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 10 / 1

L 2 discrepancy of H Σ b , m — using linear scramblings Theorem (Faure, P., Pirsic 2011) Let π ∈ S b be linear ( π ( k ) = α k (mod b )) and let Σ ∈ { π ℓ : 0 ≤ ℓ < b } m be such that � m � # { 1 ≤ i ≤ m : σ i = π ℓ } = + θ ℓ b with θ ℓ ∈ { 0 , 1 } for all 0 ≤ ℓ < b . Then we have b , m ) 2 = m ( I π, (2) − ( I π, (1) L 2 ( H Σ ) 2 ) + O (1) . b b In particular L 2 ( H Σ � b , m ) ≍ log N . Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 11 / 1

L 2 discrepancy of H Σ b , m — using linear scramblings Corollary We have � L 2 ( H Σ I π, (2) − ( I π, (1) b , m ) ) 2 b b lim √ log b m = =: c b ( π ) . log b m →∞ For example, � ( b 2 − 1)(3 b 2 + 13) c b ( id ) = . 720 b 2 log b Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 12 / 1

L 2 discrepancy of H Σ b , m — using linear scramblings � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1.6 1.4 1.2 c b � � opt � � 1.0 c b � id � 0.8 � 0.6 0.4 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0.2 0 10 20 30 40 50 Figure: Comparison of c b ( π opt ) and c b ( id ). Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 13 / 1

L 2 discrepancy of H Σ b , m — using the swapping permutation Let A ( τ b ) := { σ ∈ S b : σ ◦ τ b = τ b ◦ σ } . Theorem (Faure, P., Pirsic, Schmid 2010) Let π ∈ S b , Σ ∈ { π, τ b ◦ π } m and let ℓ = # { 1 ≤ i ≤ m : σ i = π } . Then we have b , m ) 2 = ( I π, (1) ) 2 (( m − 2 ℓ ) 2 − m ) + O ( m ) . L 2 ( H Σ b If π ∈ A ( τ b ), then � � 1 ) 2 (( m − 2 ℓ ) 2 − m ) + I π, (1) ( I π, (1) L 2 ( H Σ b , m ) 2 = 1 − (2 ℓ − m ) b b 2 b m + 3 1 1 + mI π, (2) 8 + 4 b m − 72 b 2 m . b Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 14 / 1

L 2 discrepancy of H Σ b , m — using the swapping permutation Corollary b , m ) ≍ √ log N . Choose ℓ such that ( m − 2 ℓ ) 2 = O ( m ), then L 2 ( H Σ b , m ) ≍ √ log N . Choose π ∈ S b such that I π, (1) = 0, then L 2 ( H Σ b Corollary For π ∈ A ( τ b ) we have � L 2 ( H Σ I π, (2) − ( I π, (1) b , m ) ) 2 b b lim min √ log b m = = c b ( π ) . log b m →∞ Σ ∈{ π,τ b ◦ π } m Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 15 / 1

L 2 discrepancy of H Σ b , m — using the swapping permutation b = 22 and π ∗ = (10 , 5 , 7 , 2 , 15 , 8 , 20 , 11 , 16 , 14 , 19 , 6 , 13 , 1)(4 , 18 , 17 , 3) gives � 278629 c 22 ( π ∗ ) = 2811072 log 22 = 0 . 17906 . . . . Compare: L 2 ( P ) 7 √ log N ≥ 216 √ log 2 = 0 . 038925 . . . . Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 16 / 1

L 2 discrepancy of H Σ b , m — using the swapping permutation Corollary For any σ ∈ A ( τ b ) and for any y ≥ 0 we have b ) 2 ( y 2 − 1) √ m � � � Σ ∈ { σ, τ b ◦ σ } m : L 2 ( H Σ I σ, (2) + ( I σ # b , m ) ≤ b 2 m = 2Φ( y ) − 1 + o (1) , � y −∞ e − t 2 1 where Φ( y ) = 2 d t denotes the normal distribution function. 2 π Choose Σ ∈ { σ, τ b ◦ σ } m randomly. Then, for large m , � � L 2 ( H Σ � b , m ) ≤ c log N = 1 + o (1) P for large c . Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 17 / 1