Zero-sum problems with congruence conditions

For a finite abelian group G and a positive integer d, let sdℕ(G) denote the smallest integer ℓ∈ℕ0 such that every sequence S over G of length {pipe}S{pipe}≧ℓ has a nonempty zero-sum subsequence T of length {pipe}T{pipe}≡0 mod d. We determine sdℕ(G) for all d≧1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erdo{double acute}s-Ginzburg-Ziv constant provided that, for the p-subgroups Gp of G, the Davenport constant D(Gp) is bounded above by 2exp (Gp)-1. This generalizes former results for groups of rank two. © 2011 Akadémiai Kiadó, Budapest, Hungary