Let $S(N,A,c)$ be the measure of the set of those $\alpha \in (0,1)$ such that$$\left|\alpha -\frac{x}{y}\right|<\frac{A}{y^2}$$for some $N\le y\le cN$ and $(x,y)=1$. Does$$\underset{N\to \infty }{\lim }S(N,A,c)=f(A,c)$$exist? What is its explicit form?