A Second-Logarithm Lower Bound for Sets with No Unique Sums

Abstract

For an odd prime $p$, let $m(p)$ be the minimum cardinality of a set $A\subseteq \mathbb{Z}/p\mathbb{Z}$, with $|A|\ge 2$, such that no sum in $A+A$ has a unique representation as an unordered pair from $A$, with repetition allowed. Bedert proved \[ m(p)\gg \log p\, \frac{\sqrt{\log^{(3)}p}}{\log^{(4)}p}. \] We prove the stronger lower bound \[ m(p)\gg \log p\,\log\log p. \] More generally, if $G$ is a finite Abelian group and $q(G)$ is the least prime divisor of $|G|$, then the same explicit estimate holds whenever $q(G)>2$, and in particular every subset $A\subseteq G$ with $|A|\ge 2$ and no unique sum has cardinality $\gg \log q(G)\log \log q(G)$ as $q(G)\to \infty$. The proof has two structural inputs. First, a maximum subset of $A$ whose distinct-element subset sums of size at most four are all different has cardinality $\gg \log p$. This follows from a short-coordinate lemma and a collision-lattice determinant argument. Second, we refine Bedert's density increment. Alternative representations are oriented toward an uncovered endpoint, coalesced by their translation, and separated into wide, exposed, and recurrent batches. A load-sensitive entropy lemma codes the recurrent translations using their actual final fibre multiplicities. The resulting global shift-set complexity is $\exp (O(K))$, where $K$ is the ratio of $|A|$ to the level-four additive dimension. This forces $K\gg \log \log p$, and the theorem follows. All headline statements and the structural implications used to derive them have also been checked in Lean~4 with explicit integer constants. As a secondary and logically independent result, we construct weakly ternary-balanced sets and obtain \[ m(p)\leq \frac{(\log p)^2}{2(\log 3)^2} +\left(\frac{2}{\log 3}+o(1)\right) \frac{(\log p)^2}{\log\log p}. \]