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Claim. For every $\alpha>0$ and $c\ge1$ the limit $\varrho(\alpha,c)=\lim_{m\to\infty}\mu(S(m,\alpha,c))$ exists, where $S(m,\alpha,c)$ is the set of $\xi\in[0,1]$ with $|q\xi-a|\le\alpha/q$ for some coprime $a,q$ with $m\le q\le mc$, the problem's set with $A=\alpha$, $N=m$; and

$$\varrho(\alpha,c)=\frac6{\pi^2}\sum_{r=0}^{K}(-1)^r \sum_{1\le j_1<\cdots<j_r\le K} \iint_{\mathcal H^{j_1,\ldots,j_r}(1/c)}f_{j_1,\ldots,j_r}(x,y)\,dx\,dy,$$

a finite alternating sum of double integrals of explicit functions over explicit regions of the plane, with $K$ depending on $\alpha$ and $c$ (Theorem 2, formula (10)). The authors show how this formula yields the proposers' value $12\alpha\log c/\pi^2$ for $\alpha\le c/(1+c^2)$, where $K=0$, and Kesten's closed form in the range $c^2/(1+c^2)\le\alpha c\le1$, where $K=1$. Theorem 1 adds that the limiting mass is spread uniformly: restricted to any subinterval $I\subseteq[0,1]$ the measures converge to $|I|\,\varrho(\alpha,c)$. The method relates the problem to the spacing distribution of visible lattice points under congruence constraints and uses Kloosterman sum estimates. This answers both questions of Problem 1001: the limit exists, and its form is the formula above. The paper is M. Xiong and A. Zaharescu, A problem of Erdős–Szüsz–Turán on Diophantine approximation, Acta Arith. 125 (2006), 163--177, received by the journal on 2005-10-10 and published in 2006 (the page's date is the volume's year, with the day set to its first); the card xiong_2006_problem_erdos_szusz_turan_diophantine records the paper.

Acceptance. The refereed evidence is the journal publication cited above. The reviewed evidence is the documented acceptance by the catalog erdosproblems.com, whose page for the problem (the discussion link) carries the label SOLVED and whose curator, Thomas Bloom, credits Xiong and Zaharescu, independently of Boca, with an alternative, more explicit proof of the existence of the limit; the SOLVED label is read as resting on these two proofs for the explicit form. Existence was first proved by Kesten and Sós; Boca identifies the limit independently. No independent check of the proof is recorded, and whether the formula counts as the "explicit form" Erdős asked for is a reading: it is a finite expression in integrals of elementary functions, not a closed form in $A$ and $c$ for all parameters.

Depends on. Nothing in this wiki.