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Statement

With the notation of printed p. 105 ($\varrho(P,Q)$ the distance between points of the plane, $\|x\|$ the distance from the real $x$ to the nearest integer, and $N(X,\delta)$ the maximum number of points $P_1,\ldots,P_m$ in the circle of radius $X$ with $\|\varrho(P_i,P_j)\|\ge\delta$ for $1\le i<j\le m$), as printed on pp. 106--107:

Theorem 1. "Let

$$0<\delta\le\frac1{6\cdot8^4} \tag{5}$$

and $X$ be sufficiently large depending on $\delta$. Then

$$N(X,\delta)>X^{1/2-\delta^{1/7}}. \tag{6}$$

The theorem implies obviously the

Corollary. For $\varepsilon>0$ arbitrary small, there exists $\delta_0=\delta_0(\varepsilon)>0$ such that for $0<\delta<\delta_0(\varepsilon)$, $N(X,\delta)>X^{1/2-\varepsilon}$ whenever $X$ is large enough (depending on $\varepsilon$ and $\delta$)."

Since the exponent $1/2-\delta^{1/7}$ is positive in the range (5), $N(X,\delta)\to\infty$ as $X\to\infty$ for every such $\delta$.

Source. A. Sárközy, On distances near integers, II, Studia Sci. Math. Hungar. 11 (1976), 105--111; Theorem 1 on printed p. 106 and the Corollary on p. 107 (PDF pp. 2--3 of the extract of the volume scan; volume physical pp. 112--113), read on the rendered page images (the OCR text layer garbles the formulas). The edition read is identified in the source digest.

Read depth. Claims checked: the notation, the Lemma, Theorem 1 and the Corollary were read clause by clause on the page images. The proof (pp. 107--110) was read for its structure and not checked.

Proof pointer

Pp. 107--110. Choose the integer $k$ with (7) $1/(6k^4)\ge\delta>1/(6(k+1)^4)$, so that $k\ge8$ and (9) $k>\delta^{-1/7}$, and the integer $t$ with (10) $2k^{2t+3}\le X<2k^{2(t+1)+3}$. Take all points (13) $P^{(u)}=(x^{(u)},y^{(u)})$ with $x^{(u)}=\sum_{i=0}^t\varepsilon_i^{(u)}k^{2i+2}$ and $y^{(u)}=\sum_{i=0}^t\varepsilon_i^{(u)}k^i$, the digits satisfying (14) $0\le\varepsilon_i^{(u)}\le k-2$; there are (15) $m=(k-1)^{t+1}>X^{1/2-\delta^{1/7}}$ of them and they lie in the circle of radius $X$. For $u\ne v$ the Lemma of p. 106 (if $a$ is a positive integer and $3\delta<b^2/a<2(1-\delta)$ then $\|\sqrt{a^2+b^2}\|>\delta$) is applied to $a=x^{(u)}-x^{(v)}$ and $b=y^{(u)}-y^{(v)}$, whose sizes are controlled through the highest differing digit ((19)--(28)), giving $\|\varrho(P^{(u)},P^{(v)})\|\ge\delta$ (16). Not reconstructed here.

Dependencies

The Lemma of p. 106 (proved on the same page from $a+\delta<\sqrt{a^2+b^2}<a+(1-\delta)$); otherwise self-contained.

Bears on

• Problem 466: the problem asks for some $\delta>0$ with $N(X,\delta)\to\infty$; Theorem 1 gives $N(X,\delta)>X^{1/2-\delta^{1/7}}\to\infty$ for every $\delta\le1/(6\cdot8^4)$, the site's "$N(X,\delta)>X^{1/2-\delta^{1/7}}$" for all sufficiently small $\delta$; the paper reports Graham's earlier $N(X,1/10)>\frac1{10}\log X$ on pp. 105--106.
• Problem 465: with Konyagin's $N(X,\delta)<C(\delta)X^{1/2}$ the exponent $1/2$ is the truth for small $\delta$ up to the $\delta^{1/7}$ and the constant.