Maximal (k, l)-free sets in ℤ/pℤ are arithmetic progressions

Given two different positive integers k and l, a (k, l)-free set of some group (G, +) is defined as a set  ⊂ G such that k∩l = ∅. This paper is devoted to the complete determination of the structure of (k, l)-free sets of ℤ/pℤ (p an odd prime) with maximal cardinality. Except in the case where k...

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 2002-02, Vol.65 (1), p.137-144
Main Author: Plagne, Alain
Format: Article
Language:English
Subjects:
Arithmetic
Progressions
Online Access:Get full text
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Summary:Given two different positive integers k and l, a (k, l)-free set of some group (G, +) is defined as a set  ⊂ G such that k∩l = ∅. This paper is devoted to the complete determination of the structure of (k, l)-free sets of ℤ/pℤ (p an odd prime) with maximal cardinality. Except in the case where k = 2 and l = 1 (the so-called sum-free sets), these maximal sets are shown to be arithmetic progressions. This answers affirmatively a conjecture by Bier and Chin which appeared in a recent issue of this Bulletin.
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972700020153