OP-01034 Partially resolved

The smallest set modulo a prime with no uniquely represented sum

Let $p$ be a prime. Bedert showed that the smallest size of a set $A \subset \mathbb{Z}/p\mathbb{Z}$ with $|A| \ge 2$ in which no element of $A + A$ has a unique representation lies between $\omega(p) \log p$, for some function $\omega(p) \to \infty$, and $O(\log^2 p)$, improving earlier bounds of $c \log p$ and $O(\sqrt{p})$.

Problem. Determine the size of the smallest set $A \subset \mathbb{Z}/p\mathbb{Z}$ with at least two elements such that no element of $A + A$ has a unique representation.

Source: Asked of Green by A. Granville and by S. Kopparty (2017). Listed as Problem 27 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: NT — Number Theory
Secondary: CO
Keywords: cyclic groups of prime order, sumsets, unique representations

References

@article{straus1976differences,
  author = {Straus, E. G.},
  title = {Differences of residues (mod $p$)},
  journal = {Journal of Number Theory},
  volume = {8},
  number = {1},
  pages = {40--42},
  year = {1976},
  doi = {10.1016/0022-314X(76)90019-6}
}

@article{bedert2024unique,
  author = {Bedert, Benjamin},
  title = {On unique sums in abelian groups},
  journal = {Combinatorica},
  volume = {44},
  number = {2},
  pages = {269--298},
  year = {2024},
  doi = {10.1007/s00493-023-00069-w}
}

@misc{cao2026second,
  author = {Cao, Jiao-Long and Yuan, Ye},
  title = {A second-logarithm lower bound for sets with no unique sums},
  year = {2026},
  eprint = {2608.06728},
  archivePrefix = {arXiv}
}

@misc{hou2026nearquadratic,
  author = {Hou, Jianfeng and Yang, Kai},
  title = {A near-quadratic lower bound for sets with no unique sums},
  year = {2026},
  eprint = {2610.09349},
  archivePrefix = {arXiv}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 27; version of January 2026, most recent update December 2025},
  year = {2026}
}

Resolution claims

A near-quadratic lower bound for sets with no unique sums v1 arXiv ↗ AI-Verified

Jianfeng Hou; Kai Yang

Let $m(p)$ be the least size of a subset of $\F_p$ with at least two elements for which every sum has two distinct representations as unordered pairs, allowing repetition. We prove that, for every prime $p\ge64$, \[ m(p)\ge2^{-80}\left(\frac{\log p}{\log\log p}\right)^2. \] The argument compresses the full integer collision lattice by unit-pivot elimination. A shared random sample and forests of bounded …

Partial resolution · not verified · fidelity not established · posted on arXiv 7 Oct 2026

Status history

When	Change	By	Reason
2026-10-29	Open → Partially resolved	The registry's curators	A near-quadratic lower bound for sets with no unique sums v1 recorded from arXiv:2610.09349v1.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01034, "The smallest set modulo a prime with no uniquely represented sum", clopen.solutions.