Problem 764

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claims/: The 1 claim page of Problem 764, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A\subseteq \mathbb{N}$. Can there exist some constant $c>0$ such that

$$\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+O(1)?$$

Status. Disproved, the site's label. The status-defining source is Vaughan's theorem (J. Number Theory 4 (1972), 1--16, refereed): for no set and no $c>0$ does the three-fold representation count through $N$ equal $cN$ with an error $o(N^{1/4}(\log N)^{-1/2})$, the three-summand case of a general result on $h$-fold convolutions, so a bounded error term is impossible and the answer is no. The claim page is Vaughan (accepted on the refereed publication and the site's credit), which also pins the 2026 Lean formalization of the bounded-error case in the lean-proofs repository, a development the corpus has not built.

Source. erdosproblems.com/764, accessed 2026-10-07 (empty proof-claim tab; no thread posts). Cite as: T. F. Bloom, Erdős Problem #764, https://www.erdosproblems.com/764.

References.

• [Va72] Vaughan, R. C., On the addition of sequences of integers. J. Number Theory 4 (1972), no. 1, 1-16, doi:10.1016/0022-314X(72)90008-X; not held.

Formalization. The statement is in formal-conjectures, as the existence of A : Set ℕ and c > 0 with the summatory three-fold representation count through $N$ equal to $cN$ up to $O(1)$, answer false; at its commit of 2026-09-20 the file is tagged solved and names line 3760 of src/latest/ErdosProblems/Erdos764.lean of Boris Alexeev's lean-proofs repository as the formal proof, and states Vaughan's error term as an unproved variant. That development (first added 2026-08-17; formal authors Codex and GPT-5.6 Sol) proves the bounded-error case and is pinned on the claim page. The community database (teorth/erdosproblems, file commit of 2026-09-28) records status "disproved", formal_status unformalized and formalized "yes" since 2026-09-20. The corpus has not built the development, so no formalized evidence is listed.

Progress

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Known Results

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