In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ⊂ ℝ s , we construct Kronecker-like and van der Corput-like ergodic transformations T 1,Γ and T 2,Γ of [0, 1) s . We prove that for admissible lattices Γ, (T ν,Γ n (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 ∈ L s = SL(s,ℝ)/SL(s, ℤ) (in the sense of the invariant measure on L s ), the following asymptotic formula # { 0 £ n < N:Tv,Gn(x) Î P} = NvolP + O((lnN)s + e),N ® ¥# { 0 le n < N:T_{v,Gamma }^n(x) in P} = NvolP + O({(ln N)^{s + varepsilon }}),N to infty
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机译:在本文中,我们描述了第三类低差异序列。使用晶格Γ⊂ℝ s sup>,我们构造了Kronecker型和van der Corput型的遍历变换T 1,Γ sub>和T 2,Γ sub > [0,1) s sup>。我们证明对于可允许格Γ,(t ν,Γ sub> n sup>(x))n≥0 sub>是所有x的低差异序列∈[0,1) s sup>和ν∈{1,2}。我们还证明,对于任意多面体P⊂[0,1] s sup>,对于几乎所有晶格Γ∈L s sub> = SL(s,ℝ)/ SL(s ,ℤ)(就L s sub>的不变测度而言),以下渐近公式#{0£n v,G sub> n sup>(x)ÎP} = NvolP + O((lnN) s + e sup>),N®¥#{0 le n 展开▼