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The Quarterly Journal of Mathematics Advance Access originally published online on May 28, 2007
The Quarterly Journal of Mathematics 2007 58(2):159-186; doi:10.1093/qmath/ham003
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ZERO-SUM PROBLEMS IN FINITE ABELIAN GROUPS AND AFFINE CAPS

Yves Edel1 {dagger}

Christian Elsholtz2,{ddagger}

Alfred Geroldinger3 §

Silke Kubertin4 ¶

Laurence Rackham5 ||

1 Mathematisches Institut der Universität, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany
2 Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK
3 Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria
4 Zur Seebecke 6, 31311 Uetze-Hänigsen, Germany
5 Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK

{ddagger} Corresponding author. E-mail: christian.elsholtz{at}rhul.ac.uk

Received 14 November 2006;
  Abstract

For a finite abelian group G, let Formula (G) denote the smallest integer l such that every sequence S over G of length | S| ≥ l has a zero-sum subsequence of length exp (G). We derive new upper and lower bounds for Formula (G), and all our bounds are sharp for special types of groups. The results are not restricted to groups G of the form Formula, but they respect the structure of the group. In particular, we show Formula for all odd n, which is sharp if n is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.

{dagger} E-mail: y.edel{at}mathi.uni-heidelberg.de

§ E-mail: alfred.geroldinger{at}uni-graz.at

E-mail: silke.kubertin{at}web.de

|| E-mail: L.Rackham{at}rhul.ac.uk


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