On the separating Noether number of finite abelian groups

The separating Noether number ${\beta }_{\mathrm{s}\mathrm{e}\mathrm{p}}(G)$ of a finite group $G$ is the minimal positive integer $d$ such that for every finite $G$-module $V$ there is a separating set consisting of invariant polynomials of degree at most $d$. In this paper we use methods from additive combinatorics to investigate the separating Noether number for finite abelian groups. Among others, we obtain the exact value of ${\beta }_{\mathrm{s}\mathrm{e}\mathrm{p}}(G)$, provided that $G$ is either a $p$-group or has rank $2$, $3$ or $5$.