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A solution to the Erdős–Sárközy–Sós problem on asymptotic Sidon bases of order 3
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 6 / June 2024
- Published online by Cambridge University Press:
- 10 May 2024, pp. 1418-1432
- Print publication:
- June 2024
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- Article
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A set
$S\subset {\mathbb {N}}$ is a Sidon set if all pairwise sums 
$s_1+s_2$ (for 
$s_1, s_2\in S$, 
$s_1\leqslant s_2$) are distinct. A set 
$S\subset {\mathbb {N}}$ 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 
$\mathbb {F}_q[t]$-analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.
