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

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,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 ∈ 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: T_ν,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}.