Quantitative ergodic theorems and their number-theoretic applications

TL;DR

This paper surveys recent quantitative ergodic theorems for algebraic and arithmetic group actions, highlighting their spectral method basis and diverse applications in number theory, including lattice point counting and Diophantine approximation.

Contribution

It introduces new spectral methods for deriving ergodic theorems with convergence rates and demonstrates their applications in various number-theoretic problems.

Findings

Quantitative ergodic theorems with explicit convergence rates.

Uniform error estimates in lattice point counting.

Bounds for Diophantine approximation exponents.

Abstract

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Quantitative ergodic theorems give rise to new and previously inaccessible applications, and we demonstrate the remarkable diversity of such applications by presenting several number-theoretic results. These include, in particular, general uniform error estimates in lattice points counting problems, explicit estimates in sifting problems for almost-prime points on symmetric varieties, bounds for exponents of intrinsic Diophantine approximation, and results on fast distribution of dense orbits on homogeneous spaces.