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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