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Claim. The answer to Problem 333 is no: there is a set $A\subseteq\mathbb{N}$ of density zero such that no set $B$ with $A\subseteq B+B$ satisfies $\lvert B\cap{1,\ldots,N}\rvert =o(N^{1/2})$. The result is a consequence of Theorem 2 of P. Erdős and D. J. Newman, Bases for sets of integers, J. Number Theory 9 (1977), no. 4, 420--425 (source card). For a finite set $A$ of non-negative integers let $m_A$ be the least size of a set $B$ with $A\subseteq B+B$. Theorem 2 states that most sets $A$ of $n$ elements with largest element $N$ satisfy

$$m_A>\min\!\left(\frac{n}{\log N},\ \frac{N^{1/2}}{2}\right).$$

Taking in each block of a dyadic sequence $N_1<N_2<\cdots$ a set of about $N_k^{1/2}\log N_k$ integers with largest element $N_k$ that satisfies the theorem's conclusion, and letting $A$ be the union of these pieces, gives a set of density zero; every representation of an element of the $k$-th piece as $b+b'$ uses elements of $B$, a set of non-negative integers, that are at most $N_k$, so $\lvert B\cap\{0,\ldots,N_k\}\rvert\ge m_{A_k}>N_k^{1/2}/2$ and $\lvert B\cap\{1,\ldots,N_k\}\rvert>N_k^{1/2}/2-1$ for every $k$, and the counting function of $B$ is not $o(N^{1/2})$. The forum thread's comment of 2025-12-25 sketches this deduction in one sentence, with pieces of $n=N^{0.6}$ elements glued along dyadic $N$, and the site's commentary says only that Theorem 2 implies a negative answer; the fuller argument above expands that sketch, has no independent review and rests on the statement of Theorem 2 as printed.

Acceptance. Refereed: Theorem 2 is a journal theorem (J. Number Theory). Reviewed: the deduction of the negative answer from it was pointed out in the problem's forum thread on 2025-12-25, after which the site's curator, T. F. Bloom, relabeled the problem disproved and recorded in the commentary that the theorem settles the question negatively and that Erdős and Graham seem not to have noticed this (site page last edited 2025-12-27); the curator took no part in the claim, and this is the site's acceptance, the only review listed. The preprint of Feng, Trinh, Bingham and coauthors (arXiv:2601.22401, January 2026) is the identifying team's own account, not an independent one: its Remark 5.1 says that the forum identification was posted for the team after its Gemini-based research agent Aletheia had found Theorem 2 in the literature, and the preprint lists the problem among its five literature identifications (source card). This project has not reviewed the proof of Theorem 2, and the paper's proof is not compiled.

Context. Erdős and Newman proved in the same paper (p. 423) that the first $n$ squares have a basis of at most $n/(\log n)^{M}$ elements for every fixed $M$; the site's remark credits them with the infinite positive case, a basis of the squares with counting function $o(N^{1/2})$, which is the case the problem generalizes. A separate direct construction with a Lean 4 proof is recorded at Barreto 2025. The page's date is the paper's issue month, November 1977, taken from the Crossref record; the day is the month's first.

Depends on. No page of this wiki: the claim rests on the published theorem and the deduction stated above.