A logarithmic error term for subset sums of zero-sum-free sets

Abstract

For a positive integer $k$, let $f(k)$ be the minimum number of distinct nonempty subset sums of a zero-sum-free $k$-element set, where the minimum is taken over all finite abelian groups. We prove $f(k)\ge \frac{k^2}{2}-13k\log(k+1)-116k$. Together with the classical upper bound, this gives $f(k)=k^2/2+O(k\log k)$, with the optimal leading coefficient. As a consequence, the Olson constant of every finite abelian group $G$ satisfies $\mathrm{Ol}(G)\le\sqrt{2|G|}+O(\log|G|)$. The argument combines a triangular lower bound for repeated supports, a count using blocks of two and three copies, and a greedy expansion argument. No restriction is imposed on the order of the group or on the stabilizer of the set of subset sums.