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In combinatorial number theory, zero-sum problems, subset sums and covers of the integers are three different topics initiated by P. Erd\"{o}s 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 Erd\"{o}s-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 $I\se \{1,\cdots ,k\}$ such that $\sum _{s\in I}1/n_{s}=p$ and $\sum _{s\in I}c_{s}=0$.

Copyright 2003 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 09 (2003), pp. 51-60
• Publisher Identifier: S 1079-6762(03)00111-2
• 2000 Mathematics Subject Classification. Primary 11B75; Secondary 05A05, 05C07, 11B25, 11C08, 11D68, 11P70, 11T99, 20D60
• Key words and phrases. Zero-sum, subset sums, covers of $\mathbb{Z}$
• Received by editors March 20, 2003
• Posted on July 10, 2003
• Communicated by Ronald L. Graham
• Comments (When Available)

Zhi-Wei Sun

Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

E-mail address: zwsun@nju.edu.cn

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.