134 research outputs found
A survey of some arithmetic applications of ergodic theory in negative curvature
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 , 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.Comment: 31 pages, 15 figure
On the closedness of approximation spectra
Generalizing Cusick's theorem on the closedness of the classical Lagrange
spectrum for the approximation of real numbers by rational ones, we prove that
various approximation spectra are closed, using penetration properties of the
geodesic flow in cusp neighbourhoods in negatively curved manifolds and a
result of Maucourant.Comment: Revised version. To appear in J. Theor. Nombres Bordeau
Counting arcs in negative curvature
Let M be a complete Riemannian manifold with negative curvature, and let C_-,
C_+ be two properly immersed closed convex subsets of M. We survey the
asymptotic behaviour of the number of common perpendiculars of length at most s
from C_- to C_+, giving error terms and counting with weights, starting from
the work of Huber, Herrmann, Margulis and ending with the works of the authors.
We describe the relationship with counting problems in circle packings of
Kontorovich, Oh, Shah. We survey the tools used to obtain the precise
asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We describe
several arithmetic applications, in particular the ones by the authors on the
asymptotics of the number of representations of integers by binary quadratic,
Hermitian or Hamiltonian forms.Comment: Revised version, 44 page
On the arithmetic of crossratios and generalised Mertens' formulas
We develop the relation between hyperbolic geometry and arithmetic
equidistribution problems that arises from the action of arithmetic groups on
real hyperbolic spaces, especially in dimension up to 5. We prove
generalisations of Mertens' formula for quadratic imaginary number fields and
definite quaternion algebras over the rational numbers, counting results of
quadratic irrationals with respect to two different natural complexities, and
counting results of representations of (algebraic) integers by binary
quadratic, Hermitian and Hamiltonian forms with error bounds. For each such
statement, we prove an equidistribution result of the corresponding
arithmetically defined points. Furthermore, we study the asymptotic properties
of crossratios of such points, and expand Pollicott's recent results on the
Schottky-Klein prime functions.Comment: 44 page
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