# On m-covering families of Beatty sequences with irrational moduli Artikel i vetenskaplig tidskrift, 2012

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.

Beatty sequence

Weyl criterion

## Författare

#### Peter Hegarty

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

#### Journal of Number Theory

0022-314X (ISSN) 1096-1658 (eISSN)

Vol. 132 10 2277-2296

#### Fundament

Grundläggande vetenskaper

Annan matematik

#### DOI

10.1016/j.jnt.2012.04.016

2017-10-07