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Conditions on periodicity for sum-free sets. (English) Zbl 0871.11013

A sum-free set \(S\) is ultimately periodic if there exist positive integers \(m,n_0\) such that, for \(m\geq n_0\) we have \(n\in S\) if and only if \(n+m\in S\). Based on a one-to-one correspondence between infinite binary sequences and sum-free sets P. J. Cameron [Surveys in Combinatorics 1987, Lond. Math. Soc. Lect. Note Ser. 123, 13-42 (1987; Zbl 0677.05063)] observed that the ultimate periodicity of a sum-free set implies that also the corresponding binary sequence is ultimately periodic. His question about the converse of this statement is still open.
The authors introduce two new functions defined on the positive integers in terms of which they reformulate the equivalency of the ultimate periodicity of a sum-free set and the corresponding binary sequence. In the second main result of the paper they show that if a sum-free set is not ultimately periodic then one of these new functions must grow at least logarithmically. Then the authors show how these results can be used to support the first evidence that the above question has a negative answer.

MSC:

11B83 Special sequences and polynomials
05A99 Enumerative combinatorics
11P99 Additive number theory; partitions

Citations:

Zbl 0677.05063
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References:

[1] Cameron P. J., Surveys in Combinatorics pp 13– (1987)
[2] Dickson L. E., Bull. Amer. Math. Soc. 40 pp 711– (1934) · Zbl 0010.10305
[3] Finch Steven R., Amer. Math. Monthly 99 pp 671– (1992) · Zbl 0767.11011
[4] Guy Richard K., Unsolved Problems in Number Theory (1980)
[5] DOI: 10.2307/2324219 · Zbl 0794.00001
[6] DOI: 10.1016/0097-3165(72)90083-0 · Zbl 0257.05025
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