A survey of some arithmetic applications of ergodic theory in negative curvature
Jouni Parkkonen, Frédéric Paulin
TL;DR
This survey connects arithmetic questions to the dynamics of negatively curved manifolds by developing a unified approximation framework and exploiting equidistribution and counting of common perpendiculars between convex sets in orbifolds. It demonstrates how hyperbolic dynamics and invariant measures yield concrete results on Diophantine approximation by quadratic irrationals, the distribution of rational points in real and complex settings, and arithmetically defined objects in the Heisenberg group. The work provides precise asymptotics and equidistribution statements, grounded in Patterson-Sullivan and Bowen-Margulis theory, and translates geometric ergodic properties into number-theoretic consequences with broad potential for applications in arithmetic geometry and homogeneous dynamics.
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: Géométrie et systèmes dynamiques", at the CIRM, Luminy, 2014. We thank B. Hasselblatt for his strong encouragements to write this survey.