\'Equidistribution, comptage et approximation par irrationnels quadratiques

TL;DR

This paper studies the distribution of certain geometric objects in hyperbolic manifolds and applies these results to count quadratic irrationals over various number fields, providing new asymptotic formulas.

Contribution

It establishes equidistribution of equidistant hypersurfaces in hyperbolic manifolds and derives counting results for quadratic irrationals in specific orbits.

Findings

Proves equidistribution of hypersurfaces in hyperbolic manifolds.

Provides asymptotic counts for geodesic arcs in hyperbolic manifolds.

Derives counting formulas for quadratic irrationals over number fields.

Abstract

Let $M$ be a finite volume hyperbolic manifold, we show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic on the number of geodesic arcs of lengths at most $t$, that are perpendicular to $C$ and to the boundary of a cuspidal neighbourhood of $M$. We deduce from it counting results of quadratic irrationals over $\QQ$ or over imaginary quadratic extensions of $\QQ$, in given orbits of congruence subgroups of the modular groups.