On zero-sum subsequences of cross number 1
Volume 219 / 2025
Acta Arithmetica 219 (2025), 347-363
MSC: Primary 11B30; Secondary 11P75, 20K01
DOI: 10.4064/aa240825-15-11
Published online: 6 June 2025
Abstract
For an additive finite abelian group $G$, let $S = g_1 \cdot \ldots \cdot g_l$ be a sequence over $G$ and $\mathsf k(S) = \mathrm{ord} (g_1)^{-1} + \cdots + \mathrm{ord} (g_l)^{-1}$ be its cross number. Let $\mathsf {T}(G)$ be the smallest integer $t$ such that every sequence of $t$ elements (repetition allowed) from $G$ has a zero-sum subsequence $T$ with $\mathsf {k}(T)= 1$. We study the relation of the new invariant $\mathsf {T}(G)$ to several classical invariants such as the Erdős–Ginzburg–Ziv constant $\mathsf {s}(G)$, $\eta (G)$, and $\mathsf {t}(G)$, a recently defined invariant (see the introduction). We also determine $\mathsf {T}(G)$ for some special abelian groups, including some cyclic groups.
