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 |S|≧ℓ has a nonempty zero-sum subsequence T of length |T|≡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 Erdős–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.

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