A survey of some arithmetic applications of ergodic theory in negative curvature

TL;DR

This survey explores how ergodic theory techniques applied to negatively curved manifolds yield significant results in arithmetic, including Diophantine approximation and equidistribution of arithmetically defined points.

Contribution

It compiles and explains recent advances connecting ergodic theory, geometry, and arithmetic in negatively curved spaces, highlighting new applications and methods.

Findings

Diophantine approximation of real numbers by quadratic irrationals

Equidistribution of arithmetically defined points in various spaces

Counting properties of common perpendiculars in negatively curved orbifolds

Abstract

This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in $\mathbb R$, $\mathbb C$ and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition is based on lectures at the conference "Chaire Jean Morlet: G\'eom\'etrie et syst\`emes dynamiques", at the CIRM, Luminy, 2014. We thank B. Hasselblatt for his strong encouragements to write this survey.