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ISSN 1079-6762

 

Unification of zero-sum problems, subset sums and covers of ${\mathbb Z}$


Author: Zhi-Wei Sun
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 51-60
MSC (2000): Primary 11B75; Secondary 05A05, 05C07, 11B25, 11C08, 11D68, 11P70, 11T99, 20D60
Published electronically: July 10, 2003
MathSciNet review: 1988872
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Abstract: In combinatorial number theory, zero-sum problems, subset sums and covers of the integers are three different topics initiated by P. Erdös and investigated by many researchers; they play important roles in both number theory and combinatorics. In this paper we announce some deep connections among these seemingly unrelated fascinating areas, and aim at establishing a unified theory! Our main theorem unifies many results in these three realms and also has applications in many aspects such as finite fields and graph theory. To illustrate this, here we state our extension of the Erdös-Ginzburg-Ziv theorem: If $A=\{a_{s}(\mathrm{mod} n_{s})\}_{s=1}^{k}$ covers some integers exactly $2p-1$ times and others exactly $2p$ times, where $p$ is a prime, then for any $c_{1},\cdots ,c_{k}\in \mathbb{Z} /p\mathbb{Z} $ there exists an $I\subseteq \{1,\cdots ,k\}$ such that $\sum _{s\in I}1/n_{s}=p$ and $\sum _{s\in I}c_{s}=0$.


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Additional Information

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: http://dx.doi.org/10.1090/S1079-6762-03-00111-2
PII: S 1079-6762(03)00111-2
Keywords: Zero-sum, subset sums, covers of $\mathbb{Z}$
Received by editor(s): March 20, 2003
Published electronically: July 10, 2003
Additional Notes: The website http://pweb.nju.edu.cn/zwsun/csz.htm is devoted to the topics covered by this paper.
Supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.
Dedicated: In memory of Paul Erdös
Communicated by: Ronald L. Graham
Article copyright: © Copyright 2003 American Mathematical Society