On factorizations of zero-sum sequences over abelian torsion groups I

Abstract: Let $G$ be an additive abelian group and let $G_0\subseteq G$ be a subset. A zero-sum sequence over $G_0$ is an unordered string of terms from $G_0$ (repetition of terms allowed) such that the sum of terms is $0$. The study of zero-sum sequences dates back over 60 years, and while they have often been studied for purely combinatorial interest, the original motivation was due to connections with factorization in rings of integers in algebraic number fields. In the last few decades, the connection between factorizations of zero-sum sequences and factorization of elements in rings of integers was made more precise and extended into much more general algebraic settings. This then allows the algebraic structure of factorization to be studied via combinatorial properties of zero-sum sequences. We briefly review this connection, making all notions concrete, and then turn our focus to the combinatorial part. In the past, there has been much focus on finite groups, and more recently, on subsets of finitely generated groups. However, very little was known in the case of non-finitely generated abelian groups. In part, this is because common invariants used to study factorization, such as the Davenport Constant, are no longer guaranteed to be finite. In order to better understand factorization in the setting of infinite abelian torsion groups, we introduce a new technique measuring the size of a sequence not by the number of its terms but rather by its cross number, $\sum_{i=1}^{\ell} \frac{1}{\text{\rm ord} (g_i)}$, where the $g_i\in G_0 \subseteq G$ are the terms in the sequence. Cross numbers have previously been used almost solely for finite groups. In order to adapt their use into the infinite torsion group setting, we need to introduce a new invariant, $\mathsf K_{\mathsf{inf}}(G_0)$, defined as the infimum of all cross numbers of nontrivial zero-sum sequences with terms from $G_0$. This then sets up dichotomy between when $\mathsf K_{\mathsf{inf}}(G_0)=0$ and when $\mathsf K_{\mathsf{inf}}(G_0)>0$. In this first part of two talks, we focus on when $\mathsf K_{\mathsf{inf}}(G_0)=0$, and show that factorization of zero-sum sequences can be very ill-behaved under this assumption. In the follow-up talk, we then instead consider when $\mathsf K_{\mathsf{inf}}(G_0)>0$ and see that this instead guarantees that factorization must be well-behaved, as measured by the finiteness of several commonly factorization metrics. This is joint work with David J. Grynkiewicz and Guoqing Wang.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)