On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences

Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that $\varepsilon_1 g_1 + \ldots + \varepsilon_{\ell} g_{\ell}=0$. We study sets of lengths in the monoid $\mathcal B_{\pm} (G)$ of plus-minus weighted zero-sum sequences over $G$. If $G$ is finite, then sets of lengths are highly structured. If $G$ is infinite, then every finite, nonempty subset of $\mathbb N_{\ge 2}$ is the set of lengths of some sequence $S \in \mathcal B_{\pm} (G)$.