Sarkozy 1976 distances near integers ii

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theorem_1: For 0 < delta at most 1/(6 times 8^4) and X large depending on delta, N(X, delta) exceeds X^(1/2 - delta^(1/7)); with its Corollary, for every epsilon there is delta_0 with N(X, delta) > X^(1/2 - epsilon) for all smaller delta; the power lower bound behind Problem 466.

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A. Sárközy, On distances near integers, II, Studia Scientiarum Mathematicarum Hungarica 11 (1976), 105--111 (the running head gives 105--112, the last page being blank; received February 11, 1976). The second half of the site's key Sa76 for Problems 465 and 466; Part I is filed as sarkozy_1976_distances_near_integers_i. The paper's own footnote cites Part I as "Studia Sci. Math. Hung. 10 (1975), 37--50", a misprint: Part I is in the same volume 11 (1976), pp. 37--50, as the volume scan shows.

Copy read. The copy read for this card is the paper alone: seven pages extracted from the scan of the whole volume. Provenance: the volume scan (Studia Scientiarum Mathematicarum Hungarica 11 (1976), 488 physical pages, 170,202,819 bytes) was retrieved from the REAL-J repository of the Hungarian Academy of Sciences, https://real-j.mtak.hu/5461/1/StudScientMath_11.pdf (HTTP 200, one request); the paper was extracted from it on 2026-09-18 with PyMuPDF 1.28.2 (volume pages 111--117 selected, the volume-wide structure tree, outlines, name tree and page labels detached, unreferenced objects dropped), volume physical pp. 111--117 being printed pp. 105--111 (the blank physical p. 118, printed p. 112, was not extracted); poppler's pdfseparate and pdfunite were tried first and produced files carrying the whole volume's objects, so the library extraction was used instead. The extracted file is 1,846,964 bytes, 7 pages, with an OCR text layer whose formulas are garbled; in it printed p. $n$ is PDF p. $n-104$. Every statement below was read on rendered page images. No other page of the volume is cited here. No notice is printed in the extracted pages (the first page carries only the header "Studia Scientiarum Mathematicarum Hungarica 11 (1976), 105-112"), and the repository's volume record (https://real-j.mtak.hu/5461/, read 2026-10-02) states no copyright, license or terms; the term is unstated.

Read status: claims checked for the notation and Graham's construction (printed pp. 105--106), the Lemma, Theorem 1 and the Corollary (pp. 106--107), the remark after the proof and Theorem 2 (p. 110) and the closing remarks (p. 111), each read clause by clause on the page images; the proofs of Theorems 1 and 2 (pp. 107--111) were read for their structure only and not checked. Nothing here is independently reviewed.

Contents

• Section 1 (printed p. 105): the notation of Part I; "$N(X,\delta)$ denotes the maximum of those positive integers $m$ for which there exist points $P_1,P_2,\ldots,P_m$ in the circle of radius $X$ such that $\|\varrho(P_i,P_j)\|\ge\delta$ for $1\le i<j\le n$" (the print's $n$, where $m$ is meant), with (1) $0<\delta<1/2$. Part I's result restated: $N(X,\delta)=o(X)$, "more exactly, $N(X,\delta)<\frac{4\cdot10^4}{\delta^3}\cdot\frac X{\log\log X}$ for large enough $X$". The other direction, in the paper's words (p. 105): "Erdős conjectured that for some $\delta_0>0$, $\lim_{X\to+\infty}N(X,\delta_0)=+\infty$. This conjecture has been proved by R. L. Graham (Erdős's oral communication)." The paper then reports Graham's construction (pp. 105--106), restated here: put $x_1=10$ and $x_n=2x_{n-1}$ for $n\ge2$, and let $m$ be the positive integer with $\sqrt{x_m^2+x_m^4}\le X<\sqrt{x_{m+1}^2+x_{m+1}^4}$ (the print has $x_m^4$ in the last root, a slip); take the $m$ points $P_i=(x_i,x_i^2)$, $1\le i\le m$, which lie in the disc of radius $X$ about the origin. The paper says only that "it is easy to show" that for large enough $X$ one has $m>\frac1{10}\log X$ and $\varrho(P_i,P_j)>\frac1{10}$ for $1\le i<j\le m$ (so printed, without the double bars of the norm), and concludes (2): $N(X,1/10)>\frac1{10}\log X$ for large enough $X$. Section 1 sets out to improve this bound.
• The Lemma (p. 106): for $\delta$ satisfying (1), $a$ an arbitrary positive integer and $b$ a positive number with (3) $3\delta<b^2/a<2(1-\delta)$, (4) $\|\sqrt{a^2+b^2}\|>\delta$. The paper presents this lemma as the one principle behind both Graham's construction and its own.
• Theorem 1 (p. 106): for (5) $0<\delta\le1/(6\cdot8^4)$ and $X$ sufficiently large depending on $\delta$, (6) $N(X,\delta)>X^{1/2-\delta^{1/7}}$. Corollary (p. 107): to every $\varepsilon>0$ there is a $\delta_0=\delta_0(\varepsilon)>0$ with $N(X,\delta)>X^{1/2-\varepsilon}$ for all $0<\delta<\delta_0(\varepsilon)$ and all $X$ large enough in terms of $\varepsilon$ and $\delta$.
• Proof of Theorem 1 (pp. 107--110): $k$ defined by (7) $1/(6k^4)\ge\delta>1/(6(k+1)^4)$, so $k\ge8$ and (9) $k>\delta^{-1/7}$; $t$ defined by (10) $2k^{2t+3}\le X<2k^{2(t+1)+3}$; the points (13) $P^{(u)}=(x^{(u)},y^{(u)})$ with $x^{(u)}=\sum_{i=0}^t\varepsilon_i^{(u)}k^{2i+2}$, $y^{(u)}=\sum_{i=0}^t\varepsilon_i^{(u)}k^i$, digits (14) $0\le\varepsilon_i^{(u)}\le k-2$, of number (15) $m=(k-1)^{t+1}>X^{1/2-\delta^{1/7}}$, all inside the circle of radius $X$; the Lemma applied with $a=x^{(u)}-x^{(v)}$, $b=y^{(u)}-y^{(v)}$ through the estimates (21)--(28), giving (16) $\|\varrho(P^{(u)},P^{(v)})\|\ge\delta$.
• Remark (p. 110): the paper's interest is the case $\delta\to0$, which is why the upper bound on $\delta$ in (5) is so small; the same method would raise that bound at the cost of the exponent $1/2-\delta^{1/7}$ in (6), and in particular, as the paper states it, "the right hand side of (2) can be replaced by $X^C$ (for some absolute constant $C$): $N(X,1/10)>X^C$".
• Section 2 (pp. 110--111): Erdős's other problem, "whether there exist infinitely many points $P_1,P_2,\ldots$ in the plane such that for all pairs $P_i,P_j$, $\|\varrho(P_i,P_j)\|$ is near $1/2$"; Theorem 2 (p. 110): for every $\delta>0$ some infinite set of points $P_1,P_2,\ldots$ satisfies (29) $\bigl|\|\varrho(P_i,P_j)\|-\frac12\bigr|<\delta$ for $1\le i<j$, by modifying Graham's construction ((30) $n_1=N$, $n_{k+1}=n_k^2$, $P_k=(n_k,n_k^2)$); the closing remarks that for (29) one gets $m>c_1(\delta)\log X$ points in the circle of radius $X$, sharpenable to $m>X^{c_2(\delta)}$ with $c_2(\delta)\to0$ as $\delta\to0$, and Erdős's conjecture that for (29) with $\delta=\delta(\varepsilon)$ small, "perhaps, $m<X^\varepsilon$ must hold. I have not been able to prove or disprove this conjecture." This second problem is not Problem 465 or 466.

Compiled scope

All seven pages were rendered; pp. 105--107 and 110--111 were read in full and pp. 108--109 for the structure of the proof. Theorem 1 is compiled as a statement with the proof pointer above; Graham's construction is recorded as the paper reports it, with the paper's own "it is easy to show" and no further proof; Theorem 2 is recorded in the digest only. No step was checked and nothing here is independently reviewed.

Bears on. #466: printed pp. 105--106 (PDF pp. 1--2, page images) report Erdős's conjecture $\lim_{X\to+\infty}N(X,\delta_0)=+\infty$, its proof by Graham through the points $(x_i,x_i^2)$ with $x_i=10\cdot2^{i-1}$, and the bound (2) $N(X,1/10)>\frac1{10}\log X$, the site's "Graham proved this is true"; Theorem 1 with its Corollary (pp. 106--107, PDF pp. 2--3) is the site's "$N(X,\delta)>X^{1/2-\delta^{1/7}}$" for all sufficiently small $\delta$, which gives $N(X,\delta)\to\infty$ for every $\delta\le1/(6\cdot8^4)$, and the p. 110 remark gives $N(X,1/10)>X^C$. #465: the lower bounds that show the exponent $1/2$ of Konyagin's upper bound $N(X,\delta)<C(\delta)X^{1/2}$ cannot be lowered for small $\delta$; p. 105 restates Part I's upper bound.

Results.

• Theorem 1 (p. 106): $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for $0<\delta\le1/(6\cdot8^4)$ and $X$ large depending on $\delta$; Corollary (p. 107): for every $\varepsilon>0$ there is $\delta_0(\varepsilon)$ with $N(X,\delta)>X^{1/2-\varepsilon}$ for $0<\delta<\delta_0(\varepsilon)$ and $X$ large.
• Graham's construction (pp. 105--106): $N(X,1/10)>\frac1{10}\log X$ for large $X$, reported with a sketch; the remark on p. 110 that the method of Theorem 1 gives $N(X,1/10)>X^C$ for an absolute constant $C$.
• Theorem 2 (p. 110): infinitely many points in the plane with every $\|\varrho(P_i,P_j)\|$ within $\delta$ of $1/2$, for any $\delta>0$.

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