On m-covering families of Beatty sequences with irrational moduli

TL;DR

This paper extends Uspensky's theorem to characterize eventual exact m-covers of positive integers by homogeneous and inhomogeneous Beatty sequences with irrational moduli, providing new arithmetical criteria and exploring fractional m-covers.

Contribution

It generalizes existing results to m-covers with irrational moduli and offers a new arithmetical characterization, including for inhomogeneous sequences and fractional m-values.

Findings

Generalization of Uspensky's theorem to m-covers with irrational moduli

Arithmetical characterization of inhomogeneous m-covers

Introduction of a fractional version of Beatty's theorem

Abstract

We generalise Uspensky's theorem characterising eventual exact (e.e.) covers of the positive integers by homogeneous Beatty sequences, to e.e. m-covers, for any m \in \N, by homogeneous sequences with irrational moduli. We also consider inhomogeneous sequences, again with irrational moduli, and obtain a purely arithmetical characterisation of e.e. m-covers. This generalises a result of Graham for m = 1, but when m > 1 the arithmetical description is more complicated. Finally we speculate on how one might make sense of the notion of an exact m-cover when m is not an integer, and present a "fractional version" of Beatty's theorem.