An asymptotic Sidon basis of order $3-\eta$

Abstract

Pilatte recently proved that there exists an infinite Sidon set of positive integers which is an asymptotic basis of order $3$, answering a problem posed by Erdős, Sárkőzy and Sós in 1994. In this paper, we strengthen this result by proving that for any $0<\eta <0.0527$, there exists an infinite Sidon set $\mathcal{S}\subset \mathbb{N}$ which is an asymptotic basis of order $3-\eta$; that is, every sufficiently large integer $m$ can be represented as \[ m=s_1+s_2+s_3 \] for some $s_1,s_2,s_3\in \mathcal{S}$ satisfying \[ \min\{s_1,s_2,s_3\}\leq m^{1-η}. \] To prove this, we develop a truncated version of Pilatte's construction and use a deep result of Sawin on sums of Dirichlet convolutions of the von Mangoldt function over function fields.