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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.19413 |
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| _version_ | 1866915910761054208 |
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| author | Bersudsky, Michael Shah, Nimish A. |
| author_facet | Bersudsky, Michael Shah, Nimish A. |
| contents | We study the asymptotic distribution of norm ball averages along orbits of a lattice $Γ\subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in $\mathbb{R}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone.
Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19413 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$ Bersudsky, Michael Shah, Nimish A. Dynamical Systems 37A17, 22E40 We study the asymptotic distribution of norm ball averages along orbits of a lattice $Γ\subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in $\mathbb{R}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result. |
| title | Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$ |
| topic | Dynamical Systems 37A17, 22E40 |
| url | https://arxiv.org/abs/2505.19413 |