Zero-sum problems for abelian p-groups and covers of the integers by residue classes

Abstract

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdős more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [S03b], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdős-Ginzburg-Ziv theorem in the following way: If { as(mod ns)}s=1k covers each integer either exactly 2q − 1 times or exactly 2q times where q is a prime power, then for any c1,...,ck ∈ ℤ/qℤ there exists an I ⊆ {1,...,k} such that ∑s∈I 1/ns = q and ∑s∈Ics = 0. The main theorem of this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs.