A near-quadratic lower bound for sets with no unique sums

Abstract

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 diameter give $O(\sqrt n+n/\log p)$ surviving coordinates of polynomial height for a minimal set of size $n$. A nonzero minor divisible by $p$ then gives the lower bound. We also construct weakly ternary-balanced seeds yielding \[ m(p)\le\frac{(\log p)^2}{2(\log3)^2} +\left(\frac1{4\log3}+o(1)\right) \frac{(\log p)^2}{\log\log p}. \] Consequently $m(p)=(\log p)^{2+o(1)}$ as $p$ tends to infinity through the primes. The constant-factor order of $m(p)$ remains undetermined.