Problem 1001

Let $S(N,A,c)$ be the measure of the set of those $\alpha\in (0,1)$ such that

\left\lvert \alpha-\frac{x}{y}\right\rvert< \frac{A}{y^2}

for some $N\leq y\leq cN$ and $(x,y)=1$ . Does

\lim_{N\to \infty}S(N,A,c)=f(A,c)

exist? What is its explicit form?

Source: erdosproblems.com/1001

Number theory
Diophantine approximation

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved. The site, labels the problem SOLVED, crediting the value $12A\log c/\pi^2$ for $0<A<c/(1+c^2)$ to Erdős, Szüsz and Turán [EST58], the existence of the limit to Kesten and Sós [KeSo66], and alternative, more explicit proofs of its existence to Xiong and Zaharescu [XiZa06] and to Boca [Bo08]; the SOLVED label is read as resting on those two proofs for the explicit form. The claim pages are cited in the Current assessment.

Bo08: Boca, Florin P., A problem of Erdős, Szüsz and Turán concerning Diophantine approximations. Int. J. Number Theory (2008), 691-708.
EST58: Erdős, P. and Szüsz, P. and Turán, P., Remarks on the theory of diophantine approximation. Colloq. Math. (1958), 119-126.
KeSo66: Kesten, H. and Sós, V. T., On two problems of Erdős, Szüsz and Turán concerning diophantine approximations. Acta Arith. (1966/67), 183-192.
XiZa06: Xiong, Maosheng and Zaharescu, Alexandru, A problem of Erdős-Szüsz-Turán on Diophantine approximation. Acta Arith. (2006), 163-177.

Problem 1001

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