A Unified Theory of Zero - Sum Problems , Subset Sums and Covers

Abstract. Zero-sum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some connections among these seemingly unrelated fascinating areas. In this paper we establish the surprising connections for the first time and present a further unified approach. For example, we extend the famous Erdős-Ginzburg-Ziv theorem in the following way: If {as(mod ns)}ks=1 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 ∑