Metric Diophantine approximation on homogeneous varieties

TL;DR

This paper develops a comprehensive metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups, extending classical results to multiple completions and number fields with optimal outcomes.

Contribution

It introduces a general framework for Diophantine approximation on homogeneous varieties, incorporating multiple completions and number fields, with proofs based on automorphic spectral bounds.

Findings

Results are analogous to classical Khinchin and Jarnik theorems.

Metrics are optimal in several key examples.

Establishes simultaneous approximation over various completions and number fields.

Abstract

We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using S-algebraic integers. In several important examples, the metric results we obtain are optimal. The proof uses quantitative equidistribution properties of suitable averaging operators, which are derived from spectral bounds in automorphic representations.