Generalised Beatty sets

Abstract

Generalised Beatty sets, that is, sets of the form { mα 1 + nα 2 + β : m, n ∈ N}, are studied, where ξ denotes the largest integer less than or equal to ξ. Such sets are shown to be contained in a suitable ordinary Beatty set { nα + β : n ∈ N} and equal said set save for finitely many exceptions. Moreover, bounds for the largest such exception are given.