A solution to the Erd\H{o}s-S\'ark\"ozy-S\'os problem on asymptotic Sidon bases of order 3

Abstract

A set $S\subset \mathbb{N}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leq s_2$) 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, Erd\H{o}s, S\'{a}rk\"{o}zy and S\'{o}s 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.Comment: Proof of Lemma 3.1 corrected, 15 page