A solution to the Erdős-Sárközy-Sós problem on asymptotic Sidon bases of order 3

A set $S\subset \mathbb{N}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1,s_2\in S$, $s_1\le s_2$) are distinct. A set $S\subset \mathbb{N}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the ${\mathbb{F}}_q[t]$-analogue of Montgomery’s conjecture for convolutions of the von Mangoldt function.