Let $0<ϵ<1$ and let $g_{ϵ}(N)$ be the minimal $k$ such that if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a uniformly random subset of size $k$, and$$F_A(g)=\#\left\{S\subseteq A:g=\sum _{x\in S}x\right\},$$then, with probability $\to 1$ as $N\to \infty$,$$\left|F_A(g)-\frac{2^k}{N}\right|\le ϵ\frac{2^k}{N}$$for all $g\in G$.Estimate $g_{ϵ}(N)$ - in particular, is it true that for all $ϵ>0$$$g_{ϵ}(N)=(1+o_{ϵ}(1)){\log }_{2}N?$$