Unification of zero-sum problems, subset sums and covers of ℤ

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 ( m o d n s ) } s = 1 k A=\{a_{s}(\mathrm {mod}\ n_{s})\}_{s=1}^{k} covers some integers exactly 2 p − 1 2p-1 times and others exactly 2 p 2p times, where p p is a prime, then for any c 1 , ⋯ , c k ∈ Z / p Z c_{1},\cdots ,c_{k}\in \mathbb {Z}/p\mathbb {Z} there exists an I ⊆ { 1 , ⋯ , k } I\subseteq \{1,\cdots ,k\} such that ∑ s ∈ I 1 / n s = p \sum _{s\in I}1/n_{s}=p and ∑ s ∈ I c s = 0 \sum _{s\in I}c_{s}=0 .