Dense sets without large sumsets

Abstract

We prove, for all fixed $0 < δ< 1$, and all sufficiently large $n$, that there exists $S \subset [n]$ with $|S| \ge δn$ such that $A + B \not \subset S$ for all ${A, B \subset \mathbb{N}}$ satisfying $$\min\big\{|A|, |B|\big\} \ge \big(3 + o(1)\big) \frac{\log n }{ \log (1 / δ)}.$$ A very recent result of Hernández and Hetzel shows that our bound is sharp up to a factor of 3, and together our results settle a conjecture of Kra, Moreira, Richter, and Robertson. In fact, we prove that a $δ$-dense random subset of $[n]$ is a valid choice for $S$ with high probability, and that one can take $n^{-α} \le δ\le 1 - c$ where $c > 0$ is fixed and $α> 0$ depends only on the $o(1)$ error, answering another question of the same authors in a strong form.