TY - JOUR

T1 - On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

AU - Levin, Mordechay B.

PY - 2010

Y1 - 2010

N2 - In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ⊂ ℝs, we construct Kronecker-like and van der Corput-like ergodic transformations T1,Γ and T2,Γ of [0, 1)s. We prove that for admissible lattices Γ, (Tν,Γn(x))n≥0 is a low discrepancy sequence for all x ∈ [0, 1)s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ⊂ [0, 1)s, for almost all lattices Γ ∈ Ls = SL(s,ℝ)/SL(s, ℤ) (in the sense of the invariant measure on Ls), the following asymptotic formula #{0 ≤ n < N: Tν,Γn(x) ∈ P} = N volP + O((ln N)s+e{open}, N → ∞ holds with arbitrary small e{open} > 0, for all x ∈ [0, 1)s, and ν ∈ {1, 2}.

AB - In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ⊂ ℝs, we construct Kronecker-like and van der Corput-like ergodic transformations T1,Γ and T2,Γ of [0, 1)s. We prove that for admissible lattices Γ, (Tν,Γn(x))n≥0 is a low discrepancy sequence for all x ∈ [0, 1)s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ⊂ [0, 1)s, for almost all lattices Γ ∈ Ls = SL(s,ℝ)/SL(s, ℤ) (in the sense of the invariant measure on Ls), the following asymptotic formula #{0 ≤ n < N: Tν,Γn(x) ∈ P} = N volP + O((ln N)s+e{open}, N → ∞ holds with arbitrary small e{open} > 0, for all x ∈ [0, 1)s, and ν ∈ {1, 2}.

UR - http://www.scopus.com/inward/record.url?scp=79251565237&partnerID=8YFLogxK

U2 - 10.1007/s11856-010-0058-1

DO - 10.1007/s11856-010-0058-1

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AN - SCOPUS:79251565237

SN - 0021-2172

VL - 178

SP - 61

EP - 106

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -