On factorizations of zero-sum sequences over abelian torsion groups

Abstract: Let $G$ be an additive abelian torsion 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 $0$. In the last few decades, the connection between factorizations of zero-sum sequences and factorization of elements in rings of integers has been made more precise and extended into much more general algebraic settings. The extent to which factorization are wild or well-behaved is often measured by the finiteness and size of various arithmetic factorization invariants. Some of the most common include the catenary degree $\mathsf c(G_0)$, the set of successive distances $\Delta(G_0)$, and the elastacities $\rho_k(G_0)$. We begin by introducing what these invariants are in purely combinatorial terms and explain how they measure constraint of factorization in algebraic settings.

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. The use of cross numbers allows us to define three constants, $\mathsf K(G_0)$, $\mathsf k(G_0)$ and $\mathsf K_{\mathsf{inf}}(G_0)$, defined as the supremum of all cross numbers of minimal (by inclusion) zero-sum sequences, the supremum of all cross numbers of zero-sum free sequences (sequences having no zero-sum subsequence), and the infimum of all cross numbers of nontrivial zero-sum sequences. The first two of these constants have appeared in the literature before, but the third is newly introduced here.

In the first part of this two part talk, it was shown that factorization of zero-sum sequences can be very ill-behaved when $\mathsf K_{\mathsf{inf}}(G_0)=0$. In this second part, we consider what happens when $\mathsf K_{\mathsf{inf}}(G_0)>0$, specifically in the setting of infinite abelian torsion groups with finite total rank. In this setting, the first two cross number constants $\mathsf K(G_0)$ and $\mathsf k(G_0)$ are always finite. Assuming $\delta:=\mathsf K_{\mathsf{inf}}(G_0)>0$, we then obtain a general upper bound for the catenary degree $$\mathsf c(G_0)\leq \max\{2\delta^{-1}\mathsf k(G_0)+1, \quad 2\delta^{-1}\mathsf K(G_0)\}.$$ In particular, this implies that both the set of successive distances $\Delta(G_0)$ and catenary degree are always finite under these circumstances, with explicit concrete upper bounds. Moreover, our upper bound on the catenary degree is tight, meaning there are infinite families of subsets $G_0\subseteq G$ for which equality holds above. In addition, for the special case of quasi-cyclic groups, we are able to partially characterize what subsets $G_0$ with $\mathsf K_{\mathsf{inf}}(G_0)>0$ look like and use this to give a lower bound for the elasticities $\rho_k(G_0)$. Combined with the upper bound on the catenary degree, this yields a structural description of the possible refactorization lengths of a product of $k$ irreducibles. This is joint work with Alfred Geroldinger and Guoqing Wang.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)