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

### Zhi-Wei Sun

**Abstract.**
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.

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