Polynomial effective equidistribution

Lindenstrauss, Elon; Mohammadi, Amir; Wang, Zhiren

Comptes Rendus. Mathématique, Tome 361 (2023) no. G2, pp. 507-520

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of ${\mathrm{SL}}_2\left(𝔩\right)$ in arithmetic quotients of ${\mathrm{SL}}_2\left(ℂ\right)$ and ${\mathrm{SL}}_2\left(𝔩\right)\times {\mathrm{SL}}_2\left(𝔩\right)$.

The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

Nous prouvons des théorémes d’équidistribution effectifs, avec un taux d’erreur polynomial pour les orbites des sous-groupes unipotents de ${\mathrm{SL}}_2\left(𝔩\right)$ en quotients arithmétiques de ${\mathrm{SL}}_2\left(ℂ\right)$ et ${\mathrm{SL}}_2\left(𝔩\right)\times {\mathrm{SL}}_2\left(𝔩\right)$.

La preuve est basée sur l’utilisation d’une fonction de Margulis, des outils de la géométrie d’incidence, et le trou spectral de l’espace ambiant.

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