Konyagin 2001 distances between points plane

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theorem: Konyagin's theorem that the largest set of points in a disc of radius X with all pairwise distances at least delta from every integer has fewer than C(delta) times root X points, for every X at least 1; the sharp exponent for Problem 465.

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S. V. Konyagin, О расстояниях между точками на плоскости (On the distances between points on the plane), Mat. Zametki 69 (2001), no. 4, 630--633 (Brief Communications; DOI 10.4213/mzm691; received 21 September 2000, revised 5 October 2000; in Russian). English translation: About distances between points on the plane, Math. Notes 69 (2001), no. 3--4, 578--581, DOI 10.1023/A:1010276601734 (the mathnet.ru and Crossref records read; the translation was not read, and the card cites the Russian original). The site's key Ko01 for Problem 465.

The copy read for this card is a four-page PDF of the Russian original (Ghostscript output; printed p. $n$ is PDF p. $n-629$) whose text layer is unusable, so every statement below was read on the rendered page images with the formulas as the check. The file prints "© С. В. Конягин 2001" in the footer of its first page (printed p. 630, read on the page image), and the hosting site's Terms of Use (https://www.mathnet.ru/php/agreement.phtml?option_lang=eng, read 2026-10-02) state that "All materials published on this website including full-text articles, abstracts and author indexes are fully copyrighted by Steklov Mathematical Institute, Russian Academy of Sciences, and/or by other copyright holder" and that "Reproduction or republication of the materials contained on Math-Net.Ru in any form requires written permission of the copyright holder", naming no open license, every other right reserved.

Read status: claims checked for the definitions, the introduction's attributions and the Theorem (printed p. 630, PDF p. 1), read clause by clause on the page image on 2026-09-18; the proof (pp. 630--633) was read for its structure and not checked; nothing here is independently reviewed.

Konyagin writes $\{x\}$ for the fractional part of the real number $x$ and $\|x\|$ for the distance from $x$ to the nearest integer, and $d(P,Q)$ for the distance between points of the plane; for $X>0$ and $\delta\in(0,1/2)$, $N(X,\delta)$ is the maximal number of points $P_1,\ldots,P_n$ in the disc of radius $X$ with $\|d(P_i,P_j)\|\ge\delta$ for $1\le i<j\le n$ (1). The introduction (p. 630) records the trivial bound $N(X,\delta)=O(X/\delta^2)$, uniform in $X\ge1$ and $\delta$; Erdős's proposal to strengthen it to $N(X,\delta)=o(X)$ for every $\delta$, which it says A. Sárközy [1] proved by showing $N(X,\delta)=O(X/\log\log X)$ for fixed $\delta$ and $X\ge3$; Erdős's other conjecture, $N(X,\delta)\to\infty$ as $X\to\infty$, which it says R. L. Graham proved; the best known lower bounds, due to Sárközy [2]: $N(X,\delta)>X^{c(\delta)}$ ($c(\delta)>0$) and moreover $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for $0<\delta\le1/(6\cdot8^4)$ and $X\ge X(\delta)$; and the question of Erdős and Graham [3] that the last result prompted, whether $N(X,\delta)<X^{1/2+\varepsilon}$ for all $\varepsilon>0$ and $X\ge X(\delta,\varepsilon)$, which the paper answers. The Theorem (p. 630): for every $\delta>0$ there exists $C(\delta)$ such that $N(X,\delta)<C(\delta)X^{1/2}$ for $X\ge1$. The method (pp. 630--633) is Fourier-analytic: for points $P_j=(x_j,y_j)$ the sums $A_k(\varphi)=\sum_je(kz_j(\varphi))$ with $z_j(\varphi)=x_j\cos\varphi+y_j\sin\varphi$; the nonnegativity of $\sum_kd_k\int_0^{2\pi}|A_k(\varphi)|^2d\varphi$ for nonnegative weights $d_k$ (2); the Bessel identity $\int_0^{2\pi}e(k(z_i(\varphi)-z_j(\varphi)))d\varphi=2\pi J_0(2\pi kd(P_i,P_j))$ (4), the asymptotic expansion of $J_0$ and the inequality (7) it yields (p. 631); Lemma 1 (p. 632), a cosine polynomial with nonnegative coefficients whose sum with the absolute value of its conjugate is negative on $[2\pi\delta,2\pi(1-\delta)]$; and the choice of weights $d_k=(k/2)^{1/2}c_k$ in Section 4 (p. 633), which makes the off-diagonal contribution at most $-A(2X)^{-1/2}n^2$ while the error terms are $O(n)$, giving $n=O(X^{1/2})$. The references are Sárközy's two 1976 Studia papers [1], [2], the 1980 Erdős--Graham monograph [3] and Korenev's textbook on Bessel functions [4]. The theorem settles the quantitative form of Problem 465 up to the constant $C(\delta)$.

Source: https://www.mathnet.ru/eng/mzm691.

Bears on. #465: the Theorem (printed p. 630, PDF p. 1, page image) gives $N(X,\delta)<C(\delta)X^{1/2}$ for all $X\ge1$, which answers both of the problem's questions, $N(X,\delta)=o(X)$ and $N(X,\delta)<X^{1/2+o(1)}$, for every fixed $\delta$; the introduction attributes the earlier $O(X/\log\log X)$ bound to Sárközy [1] and states the Erdős--Graham question the theorem answers. #466: the introduction (p. 630, PDF p. 1) attests that Erdős's conjecture $N(X,\delta)\to\infty$ was proved by Graham and reports Sárközy's lower bounds $N(X,\delta)>X^{c(\delta)}$ and $N(X,\delta)>X^{1/2-\delta^{1/7}}$ ($0<\delta\le1/(6\cdot8^4)$, $X\ge X(\delta)$), second-hand for that problem.

Results.

• Theorem (p. 630): for every $\delta>0$ there exists $C(\delta)$ such that $N(X,\delta)<C(\delta)X^{1/2}$ for all $X\ge1$, where $N(X,\delta)$ is the maximum number of points in the disc of radius $X$ with all pairwise distances at distance at least $\delta$ from the integers.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.