Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-15T23:13:10.008Z Has data issue: false hasContentIssue false
  • English
  • Français

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: plagne@lix.polytechnique.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Bier, T. and Chin, A.Y.M., ‘On (k, l)-sets in cyclic groups of odd prime order’, Bull. Austral. Math. Soc. 63 (2001), 115–121.CrossRefGoogle Scholar
[2]Cauchy, A.L., ‘Recherches sur les nombres’ J. École Polytech. 9 (1813), 99–123.Google Scholar
[3]Davenport, H., ‘On the addition of residue classes’, J. London Math. Soc. 10 (1935), 30–32.CrossRefGoogle Scholar
[4]Davenport, H., ‘A historical note’, J. London Math. Soc. 22 (1947), 100–101.CrossRefGoogle Scholar
[5]Hamidoune, Y.O. and Rødseth, Ø.J., ‘An inverse theorem mod p’, Acta Arith. 92 (2000), 251–262.CrossRefGoogle Scholar
[6]Mann, H.B., Addition theorems: the addition of group theory and number theory, Interscience Tracts in Pure and Applied Mathematics 18 (John Wiley, New York, London, Sydney, 1965).Google Scholar
[7]Nathanson, M.B., Additive number theory: Inverse problems and the geometry of sumsets, Graduate Texts in Mathematics 165 (Springer-Verlag, Berlin, Heidelberg, New York, 1996).CrossRefGoogle Scholar
[8]Vosper, A.G., ‘The critical pairs of subsets of a group of prime order’, J. London Math. Soc. 31 (1956), 200–205.CrossRefGoogle Scholar
[9]Vosper, A.G., ‘Addendum to: “The critical pairs of subsets of a group of prime order”’, J. London Math. Soc. 31 (1956), 280–282.CrossRefGoogle Scholar
[10]Wallis, W.D., Street, A.P. and Wallis, J.S., Combinatorics: Room squares, sum-free sets, Hadamard matrices, Lecture Notes in Mathematics 292 (Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[11]Yap, H.P., ‘Maximal sum-free sets in finite abelian groups. V’, Bull. Austral. Math. Soc. 13 (1975), 337–342.CrossRefGoogle Scholar