Diophantine approximation and automorphic spectrum

TL;DR

This paper provides quantitative bounds on how well points in homogeneous varieties of semisimple algebraic groups can be approximated by rational points, linking these bounds to the automorphic spectrum and Ramanujan conjectures.

Contribution

It introduces improved, sharp estimates for Diophantine approximation rates in homogeneous varieties, connecting them to automorphic spectral properties.

Findings

Established sharp bounds on Diophantine approximation rates

Linked approximation quality to automorphic spectrum and Ramanujan conjectures

Generalized previous estimates with improved results

Abstract

The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and is thus subject to the generalised Ramanujan conjectures.