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Claim. There is a set $A$ of positive integers such that every positive integer is $n^2+a$ with an integer $n\ge0$ and $a\in A$, and, writing $A(x)=\lvert A\cap\{1,\ldots,\lfloor x\rfloor\}\rvert$,

$$\limsup_{x\to\infty}\frac{A(x)}{\sqrt x}=2\varphi^{5/2}\approx6.66, \qquad\varphi=\frac{1+\sqrt5}{2}.$$

The set is the union over $j\ge0$ of the integers in a window starting at $\varphi^{2j}$ of length about $2\varphi^{j+1/2}$. It represents every positive integer, so it qualifies for the first question of Problem 33, and the smallest possible value of the limsup asked for there is at most $2\varphi^{5/2}$. This is Theorem 1 of Wouter van Doorn's two-page note The smallest set such that every positive integer is the sum of a square and an element from this set, in the version uploaded on 12 August 2025 (the first preprint link above). The version uploaded on 3 October 2025 (the second preprint link), the edition the source card records, narrows the windows to $[\varphi^{2j},\varphi^{2j}+2\varphi^{j+1/2}-1)$ and proves the uniform bound $A(x)<2\varphi^{5/2}\sqrt x$ for every $x>0$ (its Theorem); a remark after the proof states that the limsup of this set is still $2\varphi^{5/2}$ (the remark). The note says that it optimizes the construction Erdős gave and provides the first known improvement on the upper bound.

Submission note. Posted to the erdosproblems.com forum by Wouter van Doorn (account Woett) on 12 August 2025, in the problem's thread:

The smallest possible value of $\limsup \frac{\left|A \cap {1, 2, \ldots, N } \right|}{N^{1/2}}$ is at most $2 \varphi^{5/2} \approx 6.66$, with $\varphi$ equal to the golden ratio. I have uploaded a proof here.

Posted to the same thread by the same account on 3 October 2025, in reply to that post:

I have very marginally improved the construction to get $\left|A \cap {1, 2, \ldots, N} \right| < 2 \varphi^{5/2} \sqrt{N}$ for all $N \in \mathbb{N}$, even though the limit superior is still the same.

Covers. An upper bound of $2\varphi^{5/2}\approx6.66$ for the smallest possible limsup, the first question of Problem 33. The limsup of this set equals the bound, so the construction gives nothing smaller, and the claim does not determine the smallest limsup. The liminf question is not covered; the lower bound $4/\pi$ of Cilleruelo and Habsieger bounds the smallest limsup from below.

Claimant. Wouter van Doorn, whose erdosproblems.com profile gives that name for the account Woett, posted the bound with the note in his public GitHub repository; the note names no assistance. erdosproblems.com marked both posts as addressed by an update of the problem page, and the curator, Thomas Bloom, replied to the second with thanks. The erdosproblems.com remarks (page last edited 27 December 2025, accessed 2026-10-18) state that van Doorn has a construction with $\lvert A\cap\{1,\ldots,N\}\rvert/N^{1/2}<2\phi^{5/2}$ for all $N$.

Relation to the later bounds. Two later constructions bound the smallest limsup below this one, and both are pending: the write-up of 7 September 2026 in the same thread, with $13\sqrt2/4\approx4.596$, and Plasma AI's construction, with $9/2$.

Depends on. No page of this wiki.

Standing. Claimed. The erdosproblems.com curator credits the construction in the problem's remarks, but Bloom's catalog labeled the problem OPEN, so that remark is not acceptance. The note has no journal publication, and no Lean proof is known. The library's result pages check the theorem's proof and the limsup equality step by step against the print; that is this corpus's own reading, not an independent review. The problem's erdosproblems.com proof-claims tab holds no claim (accessed 2026-10-18).