Lax-Phillips orbit counting in higher rank
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917443572596736 |
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| author | Kontorovich, Alex Lutsko, Christopher |
| author_facet | Kontorovich, Alex Lutsko, Christopher |
| contents | Given a discrete lattice, $Γ< \operatorname{SL}_m(\mathbb{R})$, and a base point $o \in \mathbb{R}^m$, let $N_Γ(T)$ denote the number of points in the orbit $o \cdot Γ$ whose (Euclidean) length is bounded by a growing parameter, $T$. We demonstrate an abstract spectral method à la Lax-Phillips, capable of obtaining strong asymptotic estimates for $N_Γ(T)$, and compare and contrast it with other methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_07740 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lax-Phillips orbit counting in higher rank Kontorovich, Alex Lutsko, Christopher Number Theory 11H06, 11F22 Given a discrete lattice, $Γ< \operatorname{SL}_m(\mathbb{R})$, and a base point $o \in \mathbb{R}^m$, let $N_Γ(T)$ denote the number of points in the orbit $o \cdot Γ$ whose (Euclidean) length is bounded by a growing parameter, $T$. We demonstrate an abstract spectral method à la Lax-Phillips, capable of obtaining strong asymptotic estimates for $N_Γ(T)$, and compare and contrast it with other methods. |
| title | Lax-Phillips orbit counting in higher rank |
| topic | Number Theory 11H06, 11F22 |
| url | https://arxiv.org/abs/2401.07740 |