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

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by Chao Liu ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

For a positive integer kk, let f(k)f(k) be the minimum number of distinct nonempty subset sums of a zero-sum-free kk-element set, where the minimum is taken over all finite abelian groups. We prove f(k)≥k22−13klog⁡(k+1)−116kf(k)\ge \frac{k^2}{2}-13k\log(k+1)-116k. Together with the classical upper bound, this gives f(k)=k2/2+O(klog⁡k)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 GG satisfies Ol(G)≤2∣G∣+O(log⁡∣G∣)\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.

Provenance statement

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References

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Version history

  1. v1Submitted by Chao LiuInitial depositCurrentOct 28, 2026