L 2 discrepancy of digit scrambled two-dimensional Hammersley point sets

Transkript

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

2 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)} Nλ([0, t)). Definition (discrepancy) For P [0, 1) 2 the L 2 -discrepancy is defined as ( ) 1/2 L 2 (P) := P (t) 2 dt. [0,1] 2 Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 2 / 1

3 Bounds on L 2 Lower ound 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) = (Hinrichs, Markhasin, 2011). Existence result (Bilyk, Chaix, Chen, Davenport, Faure, Halton, Kritzer, Larcher, P., Pirsic, Proinov, Skriganov, Temlyakov, Roth, Schmid, White, Yu, Zarema,...) 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 scramled Hammersley point sets 3 / 1

4 2-dimensional Hammersley point sets Let N, 2. For n N 0 with define n = a 0 + a 1 + a φ (n) := a 0 + a a Hammersley point set The 2-dimensional Hammersley point set is defined as {( n ) H,m := φ (n), m : 0 n < m} where m N 0. Note: #H,m = N = m. Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 4 / 1

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

6 L 2 discrepancy of H,m Theorem (Faure, P. 2009) For any 2 and any m N 0 L 2 (H,m ) 2 = m 2 ( In particular m m. ) 2 ( ( 13 + m )) 12 2 m L 2 (H,m ) log N. Not est possile L 2 compared with Roth. Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 6 / 1

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

8 Some definitions For σ S and h {1, 2,..., 1} we define ϕ σ,h : [0, 1) R. If x [ k 1, k ), where k {1,..., }, put ϕ σ,h (x) := #{0 j < k : σ(j) < h} hx if h σ(k 1), ( h)x #{0 j < k : σ(j) h} if σ(k 1) < h. Then ( ) α H Σ,m m, β m = m j=1 ϕ σ j,h j ( β j ) where h j = h j (α, β, m). For r {1, 2} put ϕ σ,(r) := 1 h=1 ( ϕ σ,h ) r and I σ,(r) := ϕ σ,(r) (x) dx. Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 8 / 1

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

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

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

12 L 2 discrepancy of H Σ,m using linear scramlings Corollary We have For example, lim m L 2 (H,m Σ ) log m = I π,(2) c (id) = (I π,(1) ) 2 log ( 2 1)( ) log =: c (π). Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 12 / 1

13 L 2 discrepancy of H Σ,m using linear scramlings c opt c id Figure: Comparison of c (π opt ) and c (id). Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 13 / 1

14 L 2 discrepancy of H Σ,m Let A(τ ) := {σ S : σ τ = τ σ}. Theorem (Faure, P., Pirsic, Schmid 2010) using the swapping permutation Let π S, Σ {π, τ π} m and let l = #{1 i m : σ i = π}. Then we have L 2 (H,m Σ )2 = (I π,(1) ) 2 ((m 2l) 2 m) + O(m). If π A(τ ), then ( L 2 (H,m Σ )2 = (I π,(1) ) 2 ((m 2l) 2 m) + I π,(1) 1 1 ) 2 m (2l m) +mi π,(2) m m. Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 14 / 1

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

16 L 2 discrepancy of H Σ,m using the swapping permutation = 22 and π = (10, 5, 7, 2, 15, 8, 20, 11, 16, 14, 19, 6, 13, 1)(4, 18, 17, 3) gives Compare: c 22 (π ) = log 22 = L 2 (P) log N log 2 = Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 16 / 1

17 L 2 discrepancy of H Σ,m Corollary using the swapping permutation For any σ A(τ ) and for any y 0 we have { # Σ {σ, τ σ} m : L 2 (H,m Σ ) I σ,(2) + (I σ)2 (y 2 1) } m 2 m = 2Φ(y) 1 + o(1), where Φ(y) = 1 y 2π e t2 2 dt denotes the normal distriution function. Choose Σ {σ, τ σ} m randomly. Then, for large m, [ ] P L 2 (H,m Σ ) c log N = 1 + o(1) for large c. Friedrich Pillichshammer (Linz/Austria) Digit scramled Hammersley point sets 17 / 1