Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

Abstract

The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. First, we answer in the affirmative, a question raised by Kleinbock, Shi, and Weiss regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M. Schmidt which counts the number of approximates to Diophantine inequalities for a certain class of approximating functions. Further we prove “spiraling” results for the distribution of approximates of Diophantine inequalities in number fields. This generalizes the work of Athreya, Ghosh, and Tseng as well as Kleinbock, Shi, and Weiss.