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

A set S⊂ N is a Sidon set if all pairwise sums s₁+s₂ (for s₁, s₂∈ S , s₁ s₂ ) are distinct. A set S⊂ 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 Fq[t] -analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.