An application of the horocyclic flow to probabilistic number theory

Abstract: Consider the following question: Fix A > 0, c > 1 and N > 0. Choose a number x ∈ [0,1] uniformly randomly. How many rational numbers p/q exist such that |x - p/q| < A/q², where q is an integer in [N, cN]? We will use the dynamics of the horocyclic flow on the space of lattices on ℝ² to provide an answer to this question as N → ∞.