Sums of Reciprocals of Fractional Parts II

Abstract

We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of magnitude of sums of reciprocals of fractional parts and solve a conjecture posed by Beresnevich, Haynes, and Velani. The methods are based both on the geometry of numbers and on probability theory.