Smallest set in $\mathbb{Z}/p\mathbb{Z}$ with no unique sum (Green Problem 27)
Statement
Fix a prime $p$, and for $A\subseteq\mathbb{Z}/p\mathbb{Z}$ let $r_A(n)$ be the number of unordered representations $n=x+y$ with $x,y\in A$ (so $x+x$ is counted once). Say $A$ has no unique sum if no element of the sumset $A+A$ has a unique representation, i.e. $r_A(n)\ne1$ (equivalently $r_A(n)\ge2$) for every $n\in A+A$. Let $m(p)$ be the size of the smallest set $A\subseteq\mathbb{Z}/p\mathbb{Z}$ with $|A|\ge2$ and no unique sum. Determine the growth of $m(p)$.
Acceptance. ADVANCES: a certified table of exact values $m(p)$ over a range of primes (each an ILP/SAT optimum with the extremal set $A$ and an infeasibility certificate for $|A|=m(p)-1$) - new data distinguishing $\log p$ from $\log^2 p$ growth. ADVANCES: an improved construction giving $m(p)=O(\log p)$ (a structured finite witness), or a lower bound $m(p)\gg\log^{1+c}p$ by a counting / Fourier argument. FULLY RESOLVES: the correct order of magnitude of $m(p)$ (settling $\Theta(\log p)$ versus $\Theta(\log^2 p)$, or the exact asymptotic) with proof.
Background
Problem 27 in Ben Green, '100 Open Problems' (manuscript, most recent update December 2025), https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. Asked to Green by A. Granville (personal communication) and independently by Swastik Kopparty (open-problems session, Harvard 2017); not marked '(Solved)'. Best known: B. Bedert, 'On unique sums in Abelian groups', arXiv:2303.15134 (Combinatorica 44 (2024) 269-298), proved $\omega(p)\log p\le m(p)\ll(\log p)^2$ for some function $\omega(p)\to\infty$ with $p$ - significant progress, NOT a resolution (Green's Update 2023 records exactly this, improving earlier bounds $c\log p$ and $O(\sqrt p)$). Whether the truth is $\Theta(\log p)$ or $\Theta(\log^2 p)$ is unknown even conjecturally. For each fixed prime, $m(p)$ is exactly computable: minimize $|A|$ subject to 'every element of $A+A$ has $\ge2$ representations', a finite ILP / SAT / exhaustive search - a table of exact $m(p)$ values does not appear to exist. Vetted open as of 2026-07-06 (high confidence; Green's Dec-2025 manuscript lists it unresolved, and Bedert's own publication list shows no follow-up).
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Ben Green, 100 Open Problems (Dec 2025) - Problem 27 (Granville / Kopparty) | paper |
| REF-02 | Bedert, On unique sums in Abelian groups (2023) - omega(p) log p <= m(p) << (log p)^2 | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.