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Claim. For $Ac\le1$ the limit $f(A,c)=\lim_{N\to\infty}S(N,A,c)$ exists, where $S(N,A,c)$ is the measure of the $\xi\in[0,1]$ with $|b\xi-a|\le A/b$ for some coprime $a,b$ with $N\le b\le cN$ (the site's set of Problem 1001, defined with a strict inequality, differs from it by a countable set), and its value is given in closed form: for $0<A\le c/(1+c^2)$ it is $12A\log c/\pi^2$ (Theorem III of Erdős, Szüsz and Turán, which the paper cites); for $c/(1+c^2)\le A\le\min(1/2,1/c)$ it is

$$\frac{12A}{\pi^2}\log c-\frac{12}{\pi^2}\Bigl(Ac+\frac Ac-A\beta-\frac A\beta +A\Bigl(\frac1\beta-\beta\Bigr)\log\frac c\beta -\frac12\Bigl(\log\frac c\beta\Bigr)^2\Bigr), \qquad \beta=\frac{1+(1-4A^2)^{1/2}}{2A};$$

and for $1/2\le A\le1/c$ it is

$$\frac{12A}{\pi^2}\log c-\frac{12}{\pi^2}\Bigl(Ac-2A+\frac Ac-\frac12(\log c)^2\Bigr).$$

This is Theorem 2 of H. Kesten, Some probabilistic theorems on Diophantine approximations, Trans. Amer. Math. Soc. 103 (1962), no. 2, 189--217, received by the editors on 1961-07-03, presented to the Society on 1961-05-22 and published in the May 1962 issue (the page's date). The paper's main object is the limiting distribution of the smallest $\|k\xi\|$ over $1\le k\le N$ (Theorem 1), extending results of Friedman and Niven and of Erdős, Szüsz and Turán; Theorem 2 is proved by the methods of Theorem 1, and the paper notes that its upper bound for $\limsup S(N,A,c)$ is useful only when $Ac<1$. The paper has no library card; its formulas are restated, as (3) and (4), in the introduction of [[../library/irrationality/xiong_2006_problem_erdos_szusz_turan_diophantine/_index|Xiong and Zaharescu's paper]], which credits Kesten with existence in the range $Ac\le1$, and the introduction of [[../library/irrationality/kesten_1966_two_problems_erdos_szusz_turan/_index|Kesten and Sós's paper]] records that the limit had been evaluated for $A\le1/c$ before their work.

Covers. The existence and the closed form of the limit for $c/(1+c^2)\le A\le1/c$, beyond the sparse range settled by Erdős, Szüsz and Turán. It settles nothing for $Ac>1$: existence there is Kesten and Sós's, and the general form is given by Xiong and Zaharescu, whose Theorem 2 recovers these closed forms as the case $K=1$ of their formula, and by Boca.

Acceptance. The refereed evidence is the journal publication cited above. The site's page for the problem does not name this paper, so no reviewed evidence is listed. No independent check of the proof is recorded. The claim value is proved because the part it settles is proved in the affirmative, with the value found.

Depends on. Nothing in this wiki.