Erdős Problem #1001 — A finite-window Diophantine approximation distribution

Let $S(N,A,c)$ be the measure of $α∈(0,1)$ for which $|α-x/y|<A/y^2$ has a coprime solution with $N≤y≤cN$. Does $S(N,A,c)$ have a limit as $N→∞$, and what is its explicit form?