Uniform lower bound for difference bases in finite abelian groups

Determine whether there exists an absolute constant \(\epsilon>0\) such that every finite abelian group \(G\) satisfies \(\eta_1(G)>\epsilon\sqrt{|G|}\).

Background

The paper defines $\eta_1(G)$ as the minimum cardinality of a subset of an abelian group $G$ whose difference set contains every group element with at least one representation. Counting arguments provide lower bounds of square-root order, but the authors emphasize that constructions and sharp estimates depend strongly on the group. They identify the existence of a uniform positive square-root lower-bound constant across all finite abelian groups as an unresolved question cited from the literature.