一个3 - η阶的渐近西顿基
An asymptotic Sidon basis of order $3-η$
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中文总结 AI 辅助
本文强化皮拉特的结果，证明对任意\(0 < η < 0.0527\)，存在无限西顿集\(\mathcal{S}\subset \mathbb{N}\)是\(3 - η\)阶渐近基，通过发展皮拉特构造的截断版本并借助索温的深刻结果得以证明。
中文摘要 AI 辅助
皮拉特最近证明存在一个正整数的无限西顿集，它是3阶渐近基，回答了1994年埃尔德什、萨尔科齐和绍什提出的问题。本文通过证明对于任意\(0 < η < 0.0527\)，存在一个无限西顿集\(\mathcal{S}\subset \mathbb{N}\)是\(3 - η\)阶渐近基来强化此结果。即每个足够大的整数\(m\)可表示为\(m = s_1 + s_2 + s_3\)，其中\(s_1,s_2,s_3\in \mathcal{S}\)且\(\min\{s_1,s_2,s_3\}\leq m^{1 - η}\)。为证明此，我们发展了皮拉特构造的截断版本并使用了索温关于函数域上冯·曼戈尔特函数的狄利克雷卷积和的一个深刻结果。
英文摘要
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 $\mathcal{S}\subset \mathbb{N}$ which is an asymptotic basis of order $3-η$; 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.
