Central Limit Theorems for Diophantine approximants

Abstract

In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior that these functions exhibit and prove a Central Limit Theorem for them. Our approach is based on a quantitative study of higher-order correlations for functions defined on the space of lattices and a novel technique for estimating cumulants of Siegel transforms.