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Claim. The limit $f(A,c)=\lim_{N\to\infty}S(N,A,c)$ exists for all $A>0$ and $c>1$, and the paper identifies it, answering both questions of Problem 1001. The paper is F. P. Boca, A problem of Erdős, Szüsz and Turán concerning Diophantine approximations, Int. J. Number Theory 4 (2008), no. 4, 691--708, published in the August 2008 issue (the page's date is the issue month; the issue gives no day). The card boca_2008_problem_erdos_szusz_turan_diophantine_approximations records the paper and its abstract, which does not say whether the identification is a closed form or an integral expression.

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 Boca, independently of Xiong and Zaharescu, 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; the independent explicit treatment of Xiong and Zaharescu appeared two years earlier. No independent check of the proof is recorded.

Depends on. Nothing in this wiki.