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A Classification of Small MSTD Sets in Arbitrary Fields

  • Yorick Herrmann
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Abstract

A finite set A in an additive abelian group is a More Sums Than Differences (MSTD) set if |A+A| > |A−A|. We prove that no MSTD set of size $5$ exists in an additive abelian group. Using a computer program inspired by Hegarty, we then obtain classifications of MSTD sets of sizes $6$, $7$, $8$, and $9$ in arbitrary fields. We also investigate the minimal cardinality of MSTD sets that are multiplicative subgroups of Z/pZ.

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Cite

BibTeXRIS
BibTeX
@misc{herrmann2026classification,
  title = {{A Classification of Small MSTD Sets in Arbitrary Fields}},
  author = {Yorick Herrmann},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.22189},
  eprint = {2608.22189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22189},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Classification of Small MSTD Sets in Arbitrary Fields
AU  - Yorick Herrmann
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.22189
UR  - https://arxiv.org/abs/2608.22189
AB  - A finite set A in an additive abelian group is a More Sums Than Differences (MSTD) set if |A+A| > |A−A|. We prove that no MSTD set of size $5$ exists in an additive abelian group. Using a computer program inspired by Hegarty, we then obtain classifications of MSTD sets of sizes $6$, $7$, $8$, and $9$ in arbitrary fields. We also investigate the minimal cardinality of MSTD sets that are multiplicative subgroups of Z/pZ.
ER  -