Équidistribution, comptage et approximation par irrationnels quadratiques
Jouni Parkkonen, Frédéric Paulin
TL;DR
This work studies equidistribution of equidistant hypersurfaces to a finite-volume totally geodesic submanifold inside a finite-volume hyperbolic manifold and derives precise asymptotics for counting geodesic arcs perpendicular to the submanifold and a cusp boundary. The core method combines equidistribution in the unit tangent bundle with geometric volume estimates to translate geometric counts into explicit asymptotics. These results are then fed into number-theoretic problems: counting representations by binary quadratic forms, and counting quadratic irrationals (real and imaginary) within orbits of modular and Bianchi groups, with constants expressed via regulators, discriminants, zeta-values, and volumes. The paper thus builds a bridge between hyperbolic geometry and arithmetic, providing exact growth rates and effective constants for a range of classical counting problems. The methods are robust to curvature variations and accommodate congruence conditions, yielding broad applicability to Diophantine approximation in quadratic and Hermitian settings.
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.