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

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erd angstrom s 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 Erd angstrom s-Ginzburg-Ziv theorem in the following way: If { a (s) (mod n(s))} (s=1) (k) covers each integer either exactly 2q - 1 times or exactly 2q times where q is a prime power, then for any c (1),...,c (k) a a"currency sign/qa"currency sign there exists an I aS dagger {1,...,k} such that a (saI) 1/n (s) = q and a (saI) c (s) = 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.