Zero-sum problems with congruence conditions

Abstract

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum subsequence $T$ of length $|T| \equiv 0 \mod d$. We determine $\mathsf s_{d \mathbb N} (G)$ for all $d\geq 1$ 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{\H o}s--Ginzburg--Ziv constant provided that, for the $p$-subgroups $G_p$ of $G$, the Davenport constant $\mathsf D (G_p)$ is bounded above by $2 \exp (G_p)-1$. This generalizes former results for groups of rank two.