Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T15:35:31.193Z Has data issue: false hasContentIssue false
  • English
  • Français

A Weighted Generalization of Gao's n + D − 1 Theorem

Published online by Cambridge University Press:  01 November 2008

YAHYA O. HAMIDOUNE
YAHYA O. HAMIDOUNE*
Affiliation:
UPMC Université Paris 6, E. Combinatoire, Case 189, 4 Place Jussieu, 75005 Paris, France (e-mail: hamidoune@math.jussieu.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let G denote a finite abelian group of order n and Davenport constant D, and put m = n + D − 1. Let x = (x1,. . .,xm) ∈ Gm. Gao's theorem states that there is a reordering (xj1, . . ., xjm) of x such that

Let w = (x1, . . ., wm) ∈ ℤm. As a corollary of the main result, we show that there are reorderings (xj1, . . ., xjm) of x and (wk1, . . ., wkm) of w, such that where xj1 is the most repeated value in x. For w = (1, . . ., 1), this result reduces to Gao's theorem.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 6 , November 2008 , pp. 793 - 798
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Caro, Y. (1996) Zero-sum problems: A survey. Discrete Math. 152 93–113.CrossRefGoogle Scholar
[2]Erdős, P., Ginzburg, A. and Ziv, A. (1961) A theorem in additive number theory. Bull Res. Council Israel 10F 41–43.Google Scholar
[3]Erdős, P. and Heilbronn, H. (1964) On the addition of residue classes mod p. Acta Arith. 9 149–159.CrossRefGoogle Scholar
[4]Gao, W. D. (1996) A combinatorial problem on finite abelian groups. J. Number Theory 58 100–103.CrossRefGoogle Scholar
[5]Gao, W. D. and Jin, X. (2004) Weighted sums in finite cyclic groups. Discrete Math. 283 243–247.CrossRefGoogle Scholar
[6]Geroldinger, A. and Halter-Koch, F. (2006) Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Vol. 278 of Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[7]Grynkiewicz, D. J. (2006) A weighted Erdős–Ginzburg–Ziv theorem. Combinatorica 26 445–453.CrossRefGoogle Scholar
[8]Hamidoune, Y. O. (1992) On a subgroup contained in some words with a bounded length. Discrete Math. 103 171–176.Google Scholar
[9]Hamidoune, Y. O. (1995) On weighted sequence sums. Combin. Probab. Comput. 4 363–367.CrossRefGoogle Scholar
[10]Kempermann, J. H. B. (1956) On complexes in a semigroup. Nederl. Akad. Wetensch. Proc. Ser A 59, Indag. Math. 18247–254.Google Scholar
[11]Scherk, P. and Moser, L. (1955) Advanced problems and solutions: Solutions 4466. Amer. Math. Monthly 62 46–47.Google Scholar
[12]Shepherdson, J. C. (1947) On the addition of elements of a sequence. J. London Math. Soc. 22 85–88.CrossRefGoogle Scholar