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Claim. Let $F(N)$ be the least size of a set $A\subseteq\{0,1,\ldots,N\}$ with $\{0,1,\ldots,N\}\subseteq A-A$, the quantity Problem 170 asks about. B. Wichmann, A note on restricted difference bases, J. London Math. Soc. 38 (1963), no. 1, 465--466, cited as [Wi63] on the problem page, gives for all integers $r,s\ge0$ a restricted difference basis with $4r+s+3$ elements for $N=4r(r+s+2)+3(s+1)$: the Wichmann ruler whose consecutive gaps are $r$ ones, one gap $r+1$, $r$ gaps $2r+1$, $s$ gaps $4r+3$, $r+1$ gaps $2r+2$ and $r$ ones. With $s$ about $2r$ the ratio $(4r+s+3)^2/N$ tends to $3$, and the family is dense enough in $N$ that

$$\limsup_{N\to\infty}\frac{F(N)}{\sqrt N}\ \le\ \sqrt3,$$

so the limit of Problem 170 is at most $\sqrt3$, the upper bound the site's commentary credits to Wichmann. The note is not held; the bound is recorded as the site's commentary and the formal-conjectures catalog's statement file report it, with the ruler family in its usual description. The site's commentary also records that Pegg's computations ([Pe20] on the problem page) suggest $\sqrt3$ is the value of the limit; that is evidence about small $N$, not a theorem, and the value stays open.

Covers. The upper bound $\limsup_{N\to\infty}F(N)/\sqrt N\le\sqrt3$. Not covered: the value of the limit, which the problem asks for, and any lower bound.

Depends on. Nothing in this wiki: the construction is the note's own, and the existence of the limit for the restricted problem, recorded on [[problems/additive_combinatorics/E0170/claims/1948_10_30_erdos_gal|Erdős and Gál's claim page]], is not an input to the bound on its limit points.

Acceptance. Refereed: Journal of the London Mathematical Society, volume 38 (1963), no. 1, 465--466, the DOI linked above; the publication record dates the issue to 1963 without a month or day, filled to 1 January for this page's name. Reviewed is not listed: the site labels the problem OPEN, and its commentary crediting the upper bound to Wichmann is commentary on an open problem, not acceptance of a solution. Formalized is not listed: the formal-conjectures catalog states the bound, with the existence of the limit and Leech's lower bound, as the lemma erdos170.existing_bounds of its file for the problem, pinned above, and marks it research solved, but states it without a proof, and this corpus has built no proof of it.