  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 probability theory.
