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Claim. There is an infinite Sidon set $S\subset\mathbb{N}$, a set in which $a+b=c+d$ with $a,b,c,d\in S$ forces $\{a,b\}=\{c,d\}$, such that every sufficiently large integer is a sum of three elements of $S$. This answers the question of Erdős, Sárközy and Sós yes, and the order is optimal, since no Sidon set is an asymptotic basis of order two. The construction in [[../library/additive_bases/pilatte_2023_solution_erdos_sarkozy_sos_problem_asymptotic/_index|Pilatte's paper]] starts from Cilleruelo's discrete-logarithm variant of Ruzsa's dense Sidon sequence, in its $\mathbb{F}_q[t]$ form, and shows that a random choice succeeds with probability one (Theorem 5.4); purely probabilistic constructions almost surely fail to give an order-3 basis (p. 2), and the equidistribution input, which over the integers would need something like Montgomery's conjecture, comes from Sawin's theorem on the $\mathbb{F}_q[t]$ analogue of Montgomery's conjecture for convolutions of the von Mangoldt function. Earlier constructions gave Sidon asymptotic bases of order $7$, $5$ and $4$, and of every order $3+\varepsilon$ in the sense of Cilleruelo.

Depends on. No page of this wiki; the result is the paper's.

Acceptance. Refereed: C. Pilatte, A solution to the Erdős–Sárközy–Sós problem on asymptotic Sidon bases of order 3, Compositio Math. 160 (2024), no. 6, 1418–1432, published online 2024-05-10; the page name uses the date of the arXiv first version, 2023-03-16. Reviewed: the site's curator, T. F. Bloom, labels Problem 157 proved on this result at erdosproblems.com, which is the site's acceptance. The site records no formalization of the problem and formal-conjectures has no statement file for it. The linked outside Lean development (Erdos157.lean, pinned at its commit of 2026-08-25) states that Pilatte proved the theorem and cites this paper and Sawin's; its theorems erdos_157_of_sawinPrime_* follow this paper's route and each takes a hypothesis from Sawin's geometry, while its exported unconditional theorem Erdos157.erdos_157 is an earlier form of the elementary construction, with coefficients in the field of $2^{1024}$ elements, of [[problems/additive_bases/E0157/claims/2026_08_25_alexeev|the AI-generated elementary proof of 2026]], not a formalization of this paper's argument. This corpus built and audited that unconditional theorem, in a later revision of the development, and records it as formal evidence on that page; the conditional theorems prove this paper's route only under their hypotheses, so formalized is not listed here.