On the independence of Hartman sequences

Abstract

Given A ⊆ [0, 1)k and α→ = (α1,...,αk) ∈ ℝk, we define ΧA,α→ : ℤ → {0, 1} by setting ΧA,α→(n) = 1 if and only if ({α1n},..., {αkn}) ∈ A, where {α} denotes the fractional part of α, i.e. α is considered as an element of the torus ℝ/ℤ. If the topological boundary of A has Haar measure 0, then ΧA,α→, is called a Hartman sequence which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets A ⊆ [0, 1)k, B ⊆ [0, 1)l and vectors α ∈ℝk, β→ ∈ ℝ1 we have ΧA,α→ = ΧB,β→. The main tool of the proof is Weyl's theorem on uniform distribution.