Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors
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arXiv
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| Format: | Preprint |
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2026
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| author | Lindenstrauss, Elon Mohammadi, Amir Yang, Lei |
| author_facet | Lindenstrauss, Elon Mohammadi, Amir Yang, Lei |
| contents | We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ 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 $\mathrm{SL}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution.
Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors Lindenstrauss, Elon Mohammadi, Amir Yang, Lei Dynamical Systems Classical Analysis and ODEs Number Theory We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ 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 $\mathrm{SL}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution. Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions. |
| title | Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2601.09983 |