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On \(m\)-covering families of Beatty sequences with irrational moduli. (English) Zbl 1288.11028

The multiset \(S(\alpha,\beta)=\{\lfloor n\alpha+\beta\rfloor:n\in\mathbb N\}\) with \(\alpha>0\) and \(\alpha,\beta\in\mathbb R\) is called a Beatty sequence. A finite family \(\{S_i:i=1,\dots,k\}\) of Beatty sequences is called an eventual exact \(m\)-cover (\(m\)-EEC), if every sufficiently large positive integer appears exactly \(m\) times in the union of \(S_i\)’s, counting multiplicities. The author proves (a) a generalization of J. V. Uspensky’s theorem [Am. Math. Mon. 34, 516–521 (1927; JFM 53.0167.05)] characterizing the \(1\)-EEC’s of homogeneous Beatty sequences (that is those with vanishing \(\beta\)’a) to e.e. \(m\)-covers for any \(m\in\mathbb N\) with irrational \(\alpha\)’s; (b) a generalization of Th. Skolem’s [Norske Vid. Selsk. Forhdl. 30, 8 p. (1957; Zbl 0084.04402)] analogous result for inhomogeneous ones. The proofs use Weyl’s equidistribution theorem instead of Kronecker’s theorem employed by Uspensky. He also gives a negative answer to a possible generalization of R. L. Graham’s [J. Comb. Theory, Ser. A 15, 354–358 (1973; Zbl 0279.10042)] structural theorem for \(1\)-EEC’s to general \(m\)-EEC’s. Finally he speculates how one might generalize the notion of an exact \(m\)-cover (that is if every integer appears exactly \(m\) times in the union) when \(m\) is not an integer and proves a ‘fractional version’ Beatty theorem (saying that every positive integer occurs exactly once in the multiset \(S(\alpha_1,0)\cup S(\alpha_2,0)\)).

MSC:

11B83 Special sequences and polynomials
11B25 Arithmetic progressions
11B75 Other combinatorial number theory
11J71 Distribution modulo one
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References:

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