A problem of Erdös Szüsz and turán concerning diophantine approximations

For A > 0 and c > 1, let S(N, A, c) denote the set of those numbers θ ∈ ]0,1[ which satisfy θ - a/b ≤ A/b² for some coprime integers a and b with N < b ≤ cN. The problem of the existence and computation of the limit f(A, c) of the Lebesgue measure of S(N, A, c) as N → ∞ was raised by Erdös, Szüsz and Turán [3]. This limit has been shown to exist by Kesten and Sós [5] using a probabilistic argument and explicitly computed when Ac ≤ 1 by Kesten [4]. We give a complete solution proving directly the existence of this limit and identifying it in all cases.