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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 866.11019

**Autor: ** Calkin, Neil J.; Erdös, Paul

**Title: ** On a class of aperiodic sum-free sets. (In English)

**Source: ** Math. Proc. Camb. Philos. Soc. 120, No.1, 1-5 (1996).

**Review: ** A set S of positive integers is said to be sum-free if (S+S)\cap S = Ø, where S+S = **{**x+y | x,y in S**}**. A sum-free set S is complete iff it is constructed greedily from a finite set, i.e. there is an n' such that for all n > n', n in S\cup(S+S). Let us call S ultimately periodic (with respect to m) if there is an n^* such that for n > n^*, n in S <==> n+m in S. Let S(\alpha) = **{**n: **{**n\alpha**}** in (1/3,2/3)**}**. The authors note that for \alpha irrational S(\alpha) is aperiodic. The main result of the paper states: For every irrational \alpha, the set S(\alpha) is incomplete. The authors raise some open questions on this topic as well.

**Reviewer: ** N.Hegyvari (Budapest)

**Classif.: ** * 11B83 Special sequences of integers and polynomials

**Keywords: ** additive problems; aperiodic sum-free sets; aperiodic; incomplete; sum-free sets

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