[Équidistribution polynomialement effective pour les orbites unipotentes dans les produits de facteurs de $\operatorname{SL}_2$]
We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf{K}$-forms of $\operatorname{SL}_2^{n}$ where $\mathbf{K}$ is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. The key new phenomenon is the existence of many intermediate groups between the $\operatorname{SL}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstructions to equidistribution. Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions.
Nous esquissons la preuve d’un théorème d’équidistribution effectif pour les sous-groupes unipotents à un paramètre dans les quotients $S$-arithmétiques provenant de $\mathbf{K}$-formes de $\operatorname{SL}_2^{n}$, où $\mathbf{K}$ est un corps de nombres. Cela fournit une version effective des résultats d’équidistribution de Ratner et Shah, avec un taux polynomial. Le nouveau phénomène essentiel réside dans l’existence de nombreux groupes intermédiaires entre le $\operatorname{SL}_2$ contenant notre unipotent et le groupe ambiant, ce qui introduit des obstructions locales et globales potentielles à l’équidistribution. Notre approche repose sur un théorème de projection de type Bourgain en présence d’obstructions, ainsi que sur une analyse minutieuse de ces obstructions.
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Keywords: Effective equidistribution, unipotent flows, Ratner’s theorem, Bourgain projection theorem
Mots-clés : Équidistribution effective, flots unipotents, théorème de Ratner, théorème de projection de Bourgain
Elon Lindenstrauss 1 , 2 ; Amir Mohammadi 3 ; Lei Yang 4
CC-BY 4.0
Elon Lindenstrauss; Amir Mohammadi; Lei Yang. Polynomially effective equidistribution for unipotent orbits in products of $\operatorname{SL}_2$ factors. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 627-634. doi: 10.5802/crmath.848
@article{CRMATH_2026__364_G3_627_0,
author = {Elon Lindenstrauss and Amir Mohammadi and Lei Yang},
title = {Polynomially effective equidistribution for unipotent orbits in products of $\operatorname{SL}_2$ factors},
journal = {Comptes Rendus. Math\'ematique},
pages = {627--634},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.848},
language = {en},
}
TY - JOUR
AU - Elon Lindenstrauss
AU - Amir Mohammadi
AU - Lei Yang
TI - Polynomially effective equidistribution for unipotent orbits in products of $\operatorname{SL}_2$ factors
JO - Comptes Rendus. Mathématique
PY - 2026
SP - 627
EP - 634
VL - 364
PB - Académie des sciences, Paris
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%I Académie des sciences, Paris
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