A recurring problem in number theory consists in counting the number of integer solutions to certain systems of inequalities with real coefficients. Consider, for example, the system
(|qα − p| < c · T −1
and 0<q≤T ,
where α and c are fixed real numbers, and (p,q) ∈ Z^2. In Diophantine approximation one is interested in obtaining asymptotics for the number of solutions to the system (1) as T → +∞. If NT denotes the number of such solutions (p, q), then, NT = NT (α) can naturally be seen as a random variable on the interval (0, 1). It turns out that NT may be written out as the sum of ”essentially i.i.d.” random variables and thus, satisfies, in some form, the law of large numbers and the central limit theorem. This phenomenon is in fact more general and may be interpreted dynamically, by considering the limiting behavior of certain unimodular lattices in R^2 under the action of the geodesic flow. For systems depending on several parameters αi, a crucial difference arises when considering actions of one-parameter diagonal flows vs. higher-rank flows. In this talk, I will consider the more general higher-rank setup and show that a form of the central limit theorem can be obtained. I will also briefly discuss some of the Diophantine consequences of this fact. This is part of a joint work with Michael Bj ̈orklund and Alexander Gorodnik.