2026/07/13 by Wei Niu
#math.CO #math.NT
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<η<0.0527, there exists an infinite Sidon set S⊂ ℕ which is an asymptotic basis of order 3-η; that is, every sufficiently large integer m can be represented as m=s1+s2+s3 for some s1,s2,s3∈ S satisfying min\s1,s2,s3\≤ m1-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.