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Difference density and aperiodic sum-free sets. (English) Zbl 1107.11013

A set \(S\) of positive integers is called sum-free, if for all \(x,y\in S\), \(x+y\notin S\). \(S\) is called periodic, if there exists a \(p\) such that for all \(n\), \(n\in S\) if and only if \(n+p\in S\). \(S\) is called ultimately periodic, if there exists a periodic \(T\) such that \(T\) and \(S\) agree after a point. (Ultimate) periodicity of a binary sequence can be defined in an analogous way. P. Cameron [Surveys in combinatorics 1987, Pap. 11th Br. Combin. Conf., London/Engl. 1987, Lond. Math. Soc. Lect. Note Ser. 123, 13-42 (1987; Zbl 0677.05063)] introduced a natural bijection between sum-free sets and infinite binary sequences. P. Cameron observed that if a sum-free set is ultimately periodic, so is the corresponding binary sequence, and asked if the converse is true. The paper provides candidates for counterexamples, but the status of the converse is still open.

MSC:

11B75 Other combinatorial number theory

Citations:

Zbl 0677.05063
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