---

Claim. If $A\subset\mathbb{N}$ is such that every large integer is $n^2+a$ with $a\in A$ and $n\ge0$, then

$$\liminf_{N\to\infty}\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1.06,$$

which answers the second question of Problem 33 yes. The site's remarks credit the bound to L. Moser, On the additive completion of sets of integers, Proc. Sympos. Pure Math. 8, Amer. Math. Soc., Providence, R.I. (1965), 175–180, and the introductions of Habsieger's 1995 paper and of Chen's 2017 paper record it as the first affirmative answer to the question Erdős asked in 1956. The bound was later raised to $4/\pi$, independently, by Cilleruelo and Habsieger.

Covers. The liminf question (the part liminf), answered yes with the bound $1.06$. Not covered: the smallest possible limsup.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Standing. Claimed. The paper appeared in a proceedings volume (Proceedings of Symposia in Pure Mathematics 8, Theory of Numbers), for which no evidence of refereeing is recorded, so the page lists no refereed evidence; the site's curator credits the bound in the problem's remarks, but the site labels the problem OPEN, so that remark is not acceptance. The paper is not held in the library, and the statement above follows the site's remark and the two introductions cited.

Dating. The page is dated by the publication year; the volume record gives no day, and the day in the page name is a placeholder.