Konyagin 2004 problems set square free numbers

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S. V. Konyagin, Problems on the set of squarefree numbers, Izv. Math. 68 (2004), no. 3, 493--520, DOI 10.1070/IM2004v068n03ABEH000486 (printed on the first page); English translation, by V. M. Millionshchikov, of the Russian original in Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 3, 63--90, received 15 August 2003. The problem pages cite the Russian original as [Ko04] under the title "Problems of the set of square-free numbers".

The copy read for this card is the publisher's PDF of the English translation: twenty-eight letter-size pages, printed pp. 493--520 (physical p. $n$ is printed p. $492+n$), with a clean text layer; the statements below were read in the text layer and checked on the page images of pp. 494--495. The Russian original was not compared. Provenance: the copy came from a survey download set of September 2026; the download URL was not recorded. 301,610 bytes. The publisher's PDF prints "© 2004 RAS(DoM) and LMS" in the header of its first page (printed p. 493), every other right reserved.

Read status: claims checked for Theorems 1--4, whose statements were read clause by clause (pp. 494--495); the proofs (sections 2--7, pp. 496--519) were not read.

Contents

For a set $S$ of positive integers, $\mathrm{ES}_N(S)$ is the maximal cardinality of $A\subseteq\{1,\dots,N\}$ with $a+a'\in S$ for all $a,a'\in A$, the case $a=a'$ included; $S_0$ is the set of squarefree numbers and $\mathrm{ES}_N=\mathrm{ES}_N(S_0)$ (p. 493). $\mathrm{BR}_N(S)$ is the supremum of $\max_x|M(x)|/\int_0^1|M(x)|\,dx$ over nonzero trigonometric polynomials $M$ with frequencies in $S\cap[1,N]$, and $\mathrm{BR}_N=\mathrm{BR}_N(S_0)$ (pp. 493--494). $\mathrm{AP}_N$ is the maximal length of an arithmetic progression in $S_0\cap\{1,\dots,N\}$ (p. 495). $C_i$ and $c_i$ are effective positive constants.

• Prior results (pp. 493--494): Erdős and Sárközy (the paper's [6], filed as erdos_1987_divisibility_properties_integers_form) proved $(\log N)/248<\mathrm{ES}_N<3N^{3/4}\log N$ for large $N$ (1.1); Sárközy [15] (Acta Math. Hungar. 60 (1992), the problem page's [Sa92c]) improved the upper bound to $\mathrm{ES}_N\ll N^{3/4}$; Elsholtz (oral communication) noted that the method of [6] gives $\mathrm{ES}_N\gg\log N\log\log N$; Gyarmati [9] (the problem page's [Gy01]) found $A,B\subseteq\{1,\dots,N\}$ with $|A|,|B|\gg\log^2N$ and all sums $a+b$ squarefree; Balog and Ruzsa [1] proved $\mathrm{BR}_N\ll N^{3/4}\log N$ (1.2). Proposition 1 (p. 494): $\mathrm{ES}_N(S)\le\mathrm{BR}_{2N}(S)$. The author expects $\mathrm{ES}_N=o(N^\varepsilon)$ for every $\varepsilon>0$, notes $\mathrm{BR}_N\gg N^{2/3}$ by [2], and that $\mathrm{ES}_N=o(\mathrm{BR}_N)$ is unproved (p. 494).
• Theorem 1 (p. 494; proofs in section 3, pp. 499--502): there are effective positive constants $C_1,C_2$ with $\mathrm{BR}_N\le N^{11/15}\exp(C_1\log N/\sqrt{\log\log N})$ (1.3) and $\mathrm{ES}_N\le N^{11/15}\exp(C_2\log N/\sqrt{\log\log N})$ (1.4) for all $N\ge3$.
• Theorem 2 (p. 495; proof in section 2, pp. 496--499): a large sieve inequality for square moduli: for distinct positive integers $n_1,\dots,n_Z\le N$ and $Z(q,h)=|\{j:n_j\equiv h\pmod q\}|$, $\sum_{p\le X}p^2\sum_{h=0}^{p^2-1}(Z(p^2,h)-Z/p^2)^2\le C_3NZ+X^{16/5}Z^{6/5}\exp(C_3\log X/\sqrt{\log\log X})$ for all integers $X\ge3$. Through a result of Bombieri and Zannier [3] on elliptic curves it improves the Erdős--Sárközy inequality (1.5) for some $N$, $Z$ and $X$ (p. 494); for $X\le Z^{1/4}$ it gives nothing beyond (1.5) (p. 495).
• Theorem 3 (p. 495; proof in section 7, pp. 518--519, after the balanced sifting of section 6): $\mathrm{ES}_N\ge c_1\log^2N\log\log N$ for all $N\ge3$ (1.6); announced in [11] (Debrecen, 2000).
• Theorem 4 (p. 495; proof in section 5, pp. 508--510): $c_2\log^2N\le\mathrm{AP}_N\le C_4\log^2N$ for all $N\ge2$ (1.7); the lower bound gives $\mathrm{ES}_N\ge c_2\log^2N/6$ by passing to the odd terms of a coprime progression and taking every other one (p. 495).
• Section 4 (pp. 502--508) develops Brun's sieve with quadratic moduli for the proofs of Theorems 3 and 4. Not read.

Compiled scope

The introduction (pp. 493--495) was read and Theorems 1--4 are recorded as checked; the proofs were not read and nothing here is independently reviewed.

Bears on. #1109 (the problem's $f(N)$ is the paper's $\mathrm{ES}_N$, both including the doubles $2a$; Theorem 1 (1.4) gives $f(N)\le N^{11/15}\exp(C_2\log N/\sqrt{\log\log N})$, improving Sárközy's $N^{3/4}$, Theorem 3 gives $f(N)\ge c_1\log^2N\log\log N$, improving Erdős--Sárközy's $(\log N)/248$, and the expectation $\mathrm{ES}_N=o(N^\varepsilon)$ on p. 494 is the problem's first question, unproved there), #1103 (for an infinite $A$ with $A+A$ squarefree, $A\cap\{1,\dots,N\}$ is one of the sets counted by $\mathrm{ES}_N$, so (1.4) bounds its counting function by $N^{11/15}\exp(C_2\log N/\sqrt{\log\log N})$, an immediate consequence noted here and not stated in the paper; otherwise the paper treats the finite problem).

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