A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

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 R, 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.

Diophantine approximations, large intersections and geodesics in negative curvature

“…The shrinking target and spiraling problems for geodesics were considered systematically in a series of works by Hersonsky and Paulin [30][31][32][33], as well as Parkkonen and Paulin [47][48][49]. In the latter papers, the setting is quite general and includes, for example, manifolds with variable negative curvature.…”

Equidistribution and counting under equilibrium states in negative curvature and trees. Applications to non-Archimedean Diophantine approximation

“…Especially pioneered by Margulis, this field has produced a huge corpus of works, by Bowen, Cosentino, Clozel, Dani, Einseidler, Eskin, Gorodnik, Ghosh, Guivarc'h, Kim, Kleinbock, Kontorovich, Lindenstraus, Margulis, McMullen, Michel, Mohammadi, Mozes, Nevo, Oh, Pollicott, Roblin, Shah, Sharp, Sullivan, Ullmo, Weiss and the last two authors, just to mention a few contributors. We refer for now to the surveys [Bab2,Oh,PaP16,PaP17c] and we will explain in more details in this introduction the relation of our work with previous works.…”

Counting intrinsic Diophantine approximations in simple algebraic groups

“…Furthermore, the problem of Diophantine approximation on quadratic surfaces was also studied in [D05, KM15, FKMS18, AG20] using homogeneous dynamics techniques, in the form of cusp excursions of diagonalizable flows on locally symmetric spaces. Finally, we refer to the survey [PP17] regarding investigation of Diophantine approximation problems in negative curvature using geometric and ergodic-theoretic techniques.…”