On m-covering families of Beatty sequences with irrational moduli

Details On m-covering families of Beatty sequences with irrational moduli On m-covering families of Beatty sequences with irrational moduli

2010-08-13

arxiv.org/abs/1008.2295v1

Mathematics

Number Theory

21 pages

Scientific paper

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.

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