SOLVED
This has been resolved in some other way than a proof or disproof.
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?
A problem of Erdős, Szüsz, and Turán
[EST58], who proved that\[f(A,c)=\frac{12 A\log c}{\pi^2}\]when $0<A< \frac{c}{1+c^2}$, and also that if $\min(A,c)>10$ then $S(N,A,c)$ is bounded away from $0$ and $1$.
The existence of this limit was proved by Kesten and Sós
[KeSo66], without a method to determine its value. Alternative, more explicit, proofs of the existence of this limit were provided independently by Boca
[Bo08] and Xiong and Zaharescu
[XiZa06]
View the LaTeX source
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #1001, https://www.erdosproblems.com/1001, accessed 2026-01-16
