Zero-sum problems with congruence conditions

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$denotethesmallestinteger$\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $\mathrm{\mid }S\mathrm{\mid }\ge \mathrm{\ell }$ has a nonempty zero-sum subsequence $T$ of length $\mathrm{\mid }T\mathrm{\mid }\equiv 0\mathrm{m}\mathrm{o}\mathrm{d}d$. We determine $\mathsf s_{d \mathbb N} (G)$forall$d\geq 1$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.