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

  • Chao Liu
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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)≥ k2/2−13klog(k+1)−116k. Together with the classical upper bound, this gives f(k)=k2/2+O(klog k), with the optimal leading coefficient. As a consequence, the Olson constant of every finite abelian group G satisfies Ol(G)≤√(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.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026logarithmic,
  title = {{A logarithmic error term for subset sums of zero-sum-free sets}},
  author = {Chao Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38373},
  eprint = {2610.38373},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38373},
}
RIS
TY  - UNPB
TI  - A logarithmic error term for subset sums of zero-sum-free sets
AU  - Chao Liu
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38373
UR  - https://arxiv.org/abs/2610.38373
AB  - 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)≥ k^2/2−13klog(k+1)−116k. Together with the classical upper bound, this gives f(k)=k^2/2+O(klog k), with the optimal leading coefficient. As a consequence, the Olson constant of every finite abelian group G satisfies Ol(G)≤√(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.
ER  -