Best possible rates of distribution of dense lattice orbits in homogeneous spaces

TL;DR

This paper establishes optimal bounds on the rate of distribution of dense lattice orbits in homogeneous spaces, advancing Diophantine approximation by linking it to spectral properties of automorphic representations.

Contribution

It introduces a unified approach to bounding distribution rates of lattice orbits using spectral analysis and ergodic theory, achieving optimal results in Diophantine approximation.

Findings

Upper and lower bounds coincide in many cases, giving optimal approximation rates.

The bounds are explicitly related to the spectrum of the automorphic representation.

The bounds are optimal when the representation is tempered.

Abstract

The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup $\Gamma $ in a connected Lie (or algebraic) group $G$, acting on suitable homogeneous spaces $G/H$. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on $H$ and acting on $G/\Gamma$. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of $H$ in the automorphic representation on $L^2(\Gamma\setminus G)$. We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.