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Claim. For every fixed $\delta$ with $0<\delta<1/2$,

$$N(X,\delta)<\frac{4\cdot10^4}{\delta^3}\cdot\frac{X}{\log\log X}$$

once $X$ is large enough in terms of $\delta$, where $N(X,\delta)$ is the largest number of points in a disc of radius $X$ whose pairwise distances all lie at distance at least $\delta$ from the nearest integer, as Problem 465 defines it. This is the Theorem of Part I (p. 38), quoted on the result page theorem; the digest is on the source card sarkozy_1976_distances_near_integers_i. Since $X/\log\log X=o(X)$, the first question has the answer yes. The paper's p. 37 records that the conjecture reached Sárközy from Erdős orally and that Part I proves a slightly sharper form of it. The proof (Lemmas 1--5, pp. 38--50) is not checked here.

Covers. The first question of Problem 465: $N(X,\delta)=o(X)$ for every $0<\delta<1/2$. It does not cover the second question, the bound $X^{1/2+o(1)}$, since the exponent of $X$ here is $1$; both questions are answered by Konyagin's bound of order $X^{1/2}$, the accepted full claim beside this page. The lower bounds for the same quantity, Graham's construction and Sárközy's Part II, are compiled on Problem 466.

Acceptance. Refereed: A. Sárközy, On distances near integers, I, Studia Sci. Math. Hungar. 11 (1976), 37--50, received 10 December 1975. Reviewed: the site's curator, Thomas F. Bloom, marks Problem 465 PROVED and credits this paper with the first conjecture in the problem's commentary (page last edited 18 January 2026); the curator neither wrote nor submitted the result. The 1980 monograph of Erdős and Graham and Konyagin's introduction of 2001 cite the bound as proved. Not formalized. The page is dated to the year of publication, since the volume prints no fuller date for the article; the paper link leads to the repository record of the open volume scan from which the copy read was extracted.