On the separating Noether number of finite abelian groups

The separating Noether number βₛₑₚ(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 βₛₑₚ(G), provided that G is either a p-group or has rank $2$, $3$ or $5$.