Hyperplane absolute winning property of bounded orbits under diagonalizable flows on $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916330612981760 |
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| author | Sawant, Gaurav |
| author_facet | Sawant, Gaurav |
| contents | We extend the work of An, Guan and Kleinbock on bounded orbits of diagonalizable flows on $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$ to $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$, where $\mathbb{K}$ is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski's Linear Forms Theorem. We then set up an appropriate Schmidt game in $\mathbb{C}^3$ such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in $\mathbb{C}^3$ relative to an approximation by imaginary quadratic rationals in $\mathbb{K}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_16671 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hyperplane absolute winning property of bounded orbits under diagonalizable flows on $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$ Sawant, Gaurav Dynamical Systems Number Theory 11J17 We extend the work of An, Guan and Kleinbock on bounded orbits of diagonalizable flows on $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$ to $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$, where $\mathbb{K}$ is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski's Linear Forms Theorem. We then set up an appropriate Schmidt game in $\mathbb{C}^3$ such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in $\mathbb{C}^3$ relative to an approximation by imaginary quadratic rationals in $\mathbb{K}$. |
| title | Hyperplane absolute winning property of bounded orbits under diagonalizable flows on $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$ |
| topic | Dynamical Systems Number Theory 11J17 |
| url | https://arxiv.org/abs/2310.16671 |