Abstract
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 T 1,Γ and T 2,Γ of [0, 1)s. We prove that for admissible lattices Γ, (T nν,Γ (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
holds with arbitrary small ɛ > 0, for all x ∈ [0, 1)s, and ν ∈ {1, 2}.
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Levin, M.B. On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube. Isr. J. Math. 178, 61–106 (2010). https://doi.org/10.1007/s11856-010-0058-1
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DOI: https://doi.org/10.1007/s11856-010-0058-1