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arXiv · 2607.11351

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

Wei Niu
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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 Erd\H{o}s, S\'{a}rk\H{o}zy and S\'{o}s 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 \[ \min\{s_1,s_2,s_3\}\leq m^{1-\eta}. \] 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.

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Wei Niu. 2026-07-13. An asymptotic Sidon basis of order $3-\eta$. https://arxiv.org/abs/2607.11351

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