Limits of definable families and dilations in nilmanifolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917707361812480 |
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| author | Peterzil, Ya'acov Starchenko, Sergei |
| author_facet | Peterzil, Ya'acov Starchenko, Sergei |
| contents | Let $G$ be a unipotent group and $\mathcal F=\{F_t:t\in (0,\infty)\}$ a family of subsets of $G$, with $\mathcal F$ definable in an o-minimal expansion of the real field. Given a lattice $Γ\subseteq G$, we study the possible Hausdorff limits of $π(\mathcal F)$ in $G/Γ$ as $t$ tends to $\infty$ (here $π:G\to G/Γ$ is the canonical projection). Towards a solution, we associate to $\mathcal F$ finitely many real algebraic subgroups $L\subseteq G$, and, uniformly in $Γ$, determine if the only Hausdorff limit at $\infty$ is $G/Γ$, depending on whether $L^Γ=G$ or not. The special case of polynomial dilations of a definable set is treated in details. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limits of definable families and dilations in nilmanifolds Peterzil, Ya'acov Starchenko, Sergei Logic Dynamical Systems Let $G$ be a unipotent group and $\mathcal F=\{F_t:t\in (0,\infty)\}$ a family of subsets of $G$, with $\mathcal F$ definable in an o-minimal expansion of the real field. Given a lattice $Γ\subseteq G$, we study the possible Hausdorff limits of $π(\mathcal F)$ in $G/Γ$ as $t$ tends to $\infty$ (here $π:G\to G/Γ$ is the canonical projection). Towards a solution, we associate to $\mathcal F$ finitely many real algebraic subgroups $L\subseteq G$, and, uniformly in $Γ$, determine if the only Hausdorff limit at $\infty$ is $G/Γ$, depending on whether $L^Γ=G$ or not. The special case of polynomial dilations of a definable set is treated in details. |
| title | Limits of definable families and dilations in nilmanifolds |
| topic | Logic Dynamical Systems |
| url | https://arxiv.org/abs/2406.19160 |