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Statement

With the notation of printed p. 37 ($\varrho(P,Q)$ the Euclidean distance in the plane, $\|x\|$ the distance from $x$ to the nearest integer, (1) $0<\delta<1/2$, and $N(X,\delta)$ the maximal number of points $P_1,\ldots,P_n$ in the circle of radius $X$ with (2) $\|\varrho(P_i,P_j)\|\ge\delta$ for $1\le i<j\le n$), as printed on p. 38 (the paper's only theorem, unnumbered):

Theorem. "For any $\delta$ satisfying (1), we have

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

if $X$ is large enough (depending on $\delta$)."

This is a quantitative form of Erdős's conjecture (3) on p. 37, $\lim_{X\to+\infty}N(X,\delta)/X=0$ for every fixed $\delta$ satisfying (1).

Source. A. Sárközy, On distances near integers, I, Studia Sci. Math. Hungar. 11 (1976), 37--50; the Theorem on printed p. 38 (PDF p. 2 of the extract of the volume scan; volume physical p. 44), the definitions on p. 37 (PDF p. 1), 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 definitions, conjectures (3)--(4) and the Theorem were read clause by clause on the page images. The proof (pp. 38--50) was read for its structure and not checked.

Proof pointer

Pp. 38--50. Lemma 1 (p. 38) is the principle: for a line $e$ and points $Q_1,\ldots,Q_m$ with perpendicular projections $Q_i'$ on $e$, if $m>9/\delta$ and for each $i$ either $\varrho(Q_i,Q_i')<\delta/4$ or the projections satisfy $\varrho(Q_i',Q_j')>1$ and $\varrho^2(Q_i,Q_i')<\frac\delta4\varrho(Q_i',Q_j')$ for all $j\ne i$, then $\|\varrho(Q_i,Q_j)\|<\delta$ for some $i\ne j$. Lemmas 2 and 3 (Section 2) are corollaries; Lemmas 4 and 5 (Section 3) prepare the application of Lemma 3 when there are many points; Section 4 (pp. 49--50) assumes indirectly $n$ points with $n$ at least the bound, (62), and every pairwise distance at least $\delta$ from the nearest integer, (63), derives (70) and applies Lemma 3 to points $R_1,\ldots,R_v$ chosen from them, obtaining a pair with $\|\varrho(R_i,R_j)\|<\delta$, against (63). Not reconstructed here.

Dependencies

Self-contained; the paper cites no external theorem for the proof.

Bears on

• Problem 465: the first displayed question, $N(X,\delta)=o(X)$ for every $0<\delta<1/2$, is answered yes by this Theorem with an explicit rate; the second question ($N(X,\delta)<X^{1/2+o(1)}$) is answered later by Konyagin's theorem, theorem of that card, which cites this paper as its [1].