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

Published online by Cambridge University Press:  17 April 2009

Alain Plagne
Alain Plagne
Affiliation:
LIX, École polytechnique, 91128 Palaiseau Cedex, France e-mail: [email protected]
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Abstract

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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 1 , February 2002 , pp. 137 - 144
Copyright
Copyright © Australian Mathematical Society 2002

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