Zero-sum problems with congruence conditions

الأصل · EN

For a finite abelian group G and a positive integer d, let sd ₙ (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 d. We determine sd ₙ (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 Erd os--Ginzburg--Ziv constant provided that, for the p-subgroups Gₚ of G, the Davenport constant D (Gₚ) is bounded above by 2 (Gₚ)-1. This generalizes former results for groups of rank two.