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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, and let $k(n)$ be the least size of an unrestricted difference basis with respect to $n$, a set of integers whose differences represent every $1\le\nu\le n$. J. Leech, On the representation of $1,2,\ldots,n$ by differences, J. London Math. Soc. 31 (1956), no. 2, 160--169, cited as [Le56] on the problem page, sharpens the lower-bound argument of Rédei and Rényi for unrestricted bases to

$$\lim_{n\to\infty}\frac{k(n)^2}{n}\ \ge\ 2-2\inf_{\theta\ne0}\frac{\sin\theta}{\theta} \ =\ 2.4344\ldots,$$

the limit on the left existing by Rédei and Rényi. A restricted basis is in particular a basis, so $F(N)\ge k(N)$ and every limit point of $F(N)/\sqrt N$ is at least $\sqrt{2.4344\ldots}=1.5602\ldots$: the limit of Problem 170 is at least $1.56$, the lower bound the site's commentary credits to Leech. The paper is not held; the bound is recorded as the site's commentary, the formal-conjectures catalog's statement file (which states Leech's constant as $\sqrt{\sup_{\theta\ne0}2(1-\sin\theta/\theta)}$) and the abstract of Bernshteyn and Tait report it. A. Bernshteyn and M. Tait, Improved lower bound for difference bases, J. Number Theory 205 (2019), 50--58, arXiv:1901.09411, show by Fourier-analytic means that the Leech--Rédei--Rényi constant $1.5602\ldots$ is not sharp for unrestricted bases, by an unspecified $\varepsilon>0$ and with no new numerical constant; through $F(N)\ge k(N)$ the same holds for the restricted limit, so $1.56$ is the best published numerical lower bound but not the best known one. That improvement is recorded here and has no claim page of its own; the site's discussion thread (posts of 16 September 2025 and 28 July 2026) notes it.

Covers. The lower bound $\liminf_{N\to\infty}F(N)/\sqrt N\ge1.5602\ldots$. Not covered: the value of the limit, which the problem asks for, and any upper bound.

Depends on. Nothing in this wiki: the argument sharpened is Rédei and Rényi's, cited in the paper, 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 31 (1956), no. 2, 160--169, the DOI linked above; the publication record dates the issue to April 1956, filled to the first of the month for this page's name. Reviewed is not listed: the site labels the problem OPEN, and its commentary crediting the lower bound to Leech 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 Wichmann's upper 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.