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

## TL;DR

This paper establishes equidistribution results for flows on homogeneous spaces and extends Diophantine approximation results to number fields, including counting approximates and spiraling distribution properties.

## Contribution

It answers a key question on orbit equidistribution for arbitrary lattices and diagonal flows, and generalizes Diophantine approximation results to number fields.

## Key findings

- Affirmative answer to equidistribution of orbits under diagonal flows
- Number field analogue of Schmidt's approximation counting result
- Spiraling distribution results for Diophantine approximates in number fields

## Abstract

The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, 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.