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This paper is a survey of some arithmetic applications of techniques in thegeometry and ergodic theory of negatively curved Riemannian manifolds, focusingon the joint works of the authors. We describe Diophantine approximationresults of real numbers by quadratic irrational ones, and we discuss variousresults on the equidistribution in $\mathbb R$, $\mathbb C$ and in theHeisenberg groups of arithmetically defined points. We explain how theseresults are consequences of equidistribution and counting properties of commonperpendiculars between locally convex subsets in negatively curved orbifolds,proven using dynamical and ergodic properties of their geodesic flows. Thisexposition 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.

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