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

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

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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

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L2 discrepancy of digit scrambled two-dimensional Hammersley point sets

Friedrich Pillichshammer1

Linz/Austria

Joint work with Henri Faure (Marseille) Gottlieb Pirsic (Linz) Wolfgang Schmid (Salzburg)

1Supported by the Austrian Science Foundation (FWF), Project S9609. Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 1 / 1

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Discrepancy

Let P = {x0, . . . , xN−1} ⊆ [0, 1)2. For t ∈ [0, 1]2 set ∆P(t) = #{0 ≤ n < N : xn ∈ [0, t)} − Nλ([0, t)).

Definition (discrepancy)

For P ⊆ [0, 1)2 the L2-discrepancy is defined as L2(P) :=

  • [0,1]2 |∆P(t)|2 dt

1/2 .

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 2 / 1

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Bounds on L2

Lower bound on L2 (Roth 1954)

∃c > 0 such that for any P ⊆ [0, 1)2 with #P = N we have L2(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 L2(P) ≤ C

  • log N.

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 3 / 1

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2-dimensional Hammersley point sets

Let b ∈ N, b ≥ 2. For n ∈ N0 with n = a0 + a1b + a2b2 + · · · define φb(n) := a0 b + a1 b2 + a2 b3 + · · · .

Hammersley point set

The 2-dimensional Hammersley point set is defined as Hb,m :=

  • φb(n), n

bm

  • : 0 ≤ n < bm

where m ∈ N0. Note: #Hb,m = N = bm.

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 4 / 1

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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

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L2 discrepancy of Hb,m

Theorem (Faure, P. 2009)

For any b ≥ 2 and any m ∈ N0 L2(Hb,m)2 = m2 b2 − 1 12b 2 + m 3b4 + 10b2 − 13 720b2 + b2 − 1 12b

  • 1 −

1 2bm

  • + 3

8 + 1 4bm − 1 72b2m . In particular L2(Hb,m) ≍ log N. Not best possible L2 compared with Roth.

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 6 / 1

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generalized Hammersley point set

Sb := S({0, 1, . . . , b − 1}) (symmetric group); for m ∈ N0 let Σ = (σ1, . . . , σm) ∈ Sm

b ;

for 0 ≤ n < bm with n = a0 + a1b + · · · + am−1bm−1 define φΣ

b (n) := σ1(a0)

b + σ2(a1) b2 + · · · + σm(am−1) bm .

generalized Hammersley point set

Let m ∈ N0 and let Σ ∈ Sm

b . The generalized 2-dimensional

Hammersley point set is defined as H Σ

b,m :=

  • φΣ

b (n), n

bm

  • : 0 ≤ n < bm

. If σi ≡ id we obtain Hb,m. Obviously: #H Σ

b,m = N = bm

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 7 / 1

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Some definitions

For σ ∈ Sb and h ∈ {1, 2, . . . , b − 1} we define ϕσ

b,h : [0, 1) → R.

If x ∈ k−1

b , k b

  • , where k ∈ {1, . . . , b}, put

ϕσ

b,h(x) :=

   #{0 ≤ j < k : σ(j) < h} − hx if h ≤ σ(k − 1), (b − h)x − #{0 ≤ j < k : σ(j) ≥ h} if σ(k − 1) < h. Then ∆H Σ

b,m

α bm , β bm

  • =

m

  • j=1

ϕσj

b,hj

β bj

  • where hj = hj(α, β, m).

For r ∈ {1, 2} put ϕσ,(r)

b

:=

b−1

  • h=1
  • ϕσ

b,h

r and I σ,(r)

b

:= 1 b 1 ϕσ,(r)

b

(x) dx.

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 8 / 1

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Some definitions

Example: b = 2, σ = (0, 1) ∈ S2 and h = 1: x ∈

  • 0, 1

2

  • , i.e., k = 1 and σ(k − 1) = σ(0) = 1. Hence

ϕσ

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

ϕσ

2,1(x) = x − #{0 ≤ j < 2 : σ(j) ≥ 1} = x − 1.

Hence ϕσ,(1)

2

(x) = ϕσ

2,1(x) = − min(x, 1 − x) =: −x

and I σ,(1)

2

= −1 8.

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 9 / 1

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Some definitions

For π ∈ Sb define πℓ(k) := π(k) + ℓ (mod b) (linear scrambling). We consider Σ ∈ {πℓ : 0 ≤ ℓ < b}m. White (1975): Σ = (id0, id1, . . . , idb−1, id0, id1, . . . , idb−1, . . .). Let τb ∈ Sb, τb(k) := b − 1 − k (swapping permutation). For π ∈ Sb we consider Σ ∈ {π, τb ◦ π}m. Note that for b = 2, π = id we have id1 = τ2 (Halton & Zaremba (1969), Kritzer & P. (2006, 2007)).

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 10 / 1

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L2 discrepancy of H Σ

b,m — using linear scramblings

Theorem (Faure, P., Pirsic 2011)

Let π ∈ Sb be linear (π(k) = αk (mod b)) and let Σ ∈ {πℓ : 0 ≤ ℓ < b}m be such that #{1 ≤ i ≤ m : σi = πℓ} = m b

  • + θℓ

with θℓ ∈ {0, 1} for all 0 ≤ ℓ < b. Then we have L2(HΣ

b,m)2 = m(I π,(2) b

− (I π,(1)

b

)2) + O(1). In particular L2(HΣ

b,m) ≍

  • log N.

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L2 discrepancy of H Σ

b,m — using linear scramblings

Corollary

We have lim

m→∞

L2(H Σ

b,m)

√log bm =

  • I π,(2)

b

− (I π,(1)

b

)2 log b =: cb(π). For example, cb(id) =

  • (b2 − 1)(3b2 + 13)

720b2 log b .

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 12 / 1

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L2 discrepancy of H Σ

b,m — using linear scramblings

  • 10

20 30 40 50 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

  • cbid
  • cbopt

Figure: Comparison of cb(πopt) and cb(id).

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 13 / 1

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L2 discrepancy of H Σ

b,m — using the swapping permutation

Let A(τb) := {σ ∈ Sb : σ ◦ τb = τb ◦ σ}.

Theorem (Faure, P., Pirsic, Schmid 2010)

Let π ∈ Sb, Σ ∈ {π, τb ◦ π}m and let ℓ = #{1 ≤ i ≤ m : σi = π}. Then we have L2(H Σ

b,m)2 = (I π,(1) b

)2((m − 2ℓ)2 − m) + O(m). If π ∈ A(τb), then L2(H Σ

b,m)2

= (I π,(1)

b

)2((m − 2ℓ)2 − m) + I π,(1)

b

  • 1 −

1 2bm

  • (2ℓ − m)

+mI π,(2)

b

+ 3 8 + 1 4bm − 1 72b2m .

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 14 / 1

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L2 discrepancy of H Σ

b,m — using the swapping permutation

Corollary

Choose ℓ such that (m − 2ℓ)2 = O(m), then L2(H Σ

b,m) ≍ √log N.

Choose π ∈ Sb such that I π,(1)

b

= 0, then L2(H Σ

b,m) ≍ √log N.

Corollary

For π ∈ A(τb) we have lim

m→∞

min

Σ∈{π,τb◦π}m

L2(H Σ

b,m)

√log bm =

  • I π,(2)

b

− (I π,(1)

b

)2 log b = cb(π).

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 15 / 1

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L2 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 c22(π∗) =

  • 278629

2811072 log 22 = 0.17906 . . . . Compare: L2(P) √log N ≥ 7 216√log 2 = 0.038925 . . . .

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 16 / 1

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L2 discrepancy of H Σ

b,m — using the swapping permutation

Corollary

For any σ ∈ A(τb) and for any y ≥ 0 we have #

  • Σ ∈ {σ, τb ◦ σ}m : L2(H Σ

b,m) ≤

  • I σ,(2)

b

+ (I σ

b )2(y2 − 1)√m

  • 2m

= 2Φ(y) − 1 + o(1), where Φ(y) =

1 2π

y

−∞ e− t2

2 dt denotes the normal distribution function.

Choose Σ ∈ {σ, τb ◦ σ}m randomly. Then, for large m, P

  • L2(H Σ

b,m) ≤ c

  • log N
  • = 1 + o(1)

for large c.

Friedrich Pillichshammer (Linz/Austria) Digit scrambled Hammersley point sets 17 / 1