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

Peter Hegarty (Institutionen för matematiska vetenskaper, matematik)
Journal of Number Theory (0022-314X). Vol. 132 (2012), 10, p. 2277-2296.

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.

Nyckelord: Beatty sequence, Weyl criterion

CPL Pubid: 159181

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# Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)