A sparse equidistribution result for $(\operatorname {SL}(2,\mathbb {R})/\Gamma _0)^{n}$
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Abstract:
Let $G=\operatorname {SL}(2,\mathbb {R})^n$, let $\Gamma =\Gamma _0^n$, where $\Gamma _0$ is a co-compact lattice in $\operatorname {SL}(2,\mathbb {R})$, let $F(\mathbf {x})$ be a non-singular quadratic form and let $u(x_1,\dots ,x_n)≔{\bigl (\begin {smallmatrix} 1 & x_1 \\ 0 & 1 \end {smallmatrix}\bigr )} \times \dots \times {\bigl (\begin {smallmatrix} 1 & x_n \\ 0 & 1 \end {smallmatrix}\bigr )}$ denote unipotent elements in $G$ which generate an $n$ dimensional horospherical subgroup. We prove that in the absence of any local obstructions for $F$, given any $x_0\in G/\Gamma$, the sparse subset $\{u(\mathbf {x})x_0:\mathbf {x}\in \mathbb {Z}^n, F(\mathbf {x})=0\}$ equidistributes in $G/\Gamma$ as long as $n\geq 481$, independent of the spectral gap of $\Gamma _0$.References
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Additional Information
- Pankaj Vishe
- Affiliation: Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom
- MR Author ID: 1005529
- ORCID: 0000-0002-7355-0615
- Email: pankaj.vishe@durham.ac.uk
- Received by editor(s): September 28, 2020
- Received by editor(s) in revised form: March 22, 2021, and June 2, 2021
- Published electronically: November 5, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 669-694
- MSC (2020): Primary 37D40; Secondary 11D72, 37A25
- DOI: https://doi.org/10.1090/tran/8498
- MathSciNet review: 4358679