Multiplicities in the length spectrum and growth rate of Salem numbers
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909237151531008 |
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| author | Grebennikov, Alexandr |
| author_facet | Grebennikov, Alexandr |
| contents | We prove that mean multiplicities in the length spectrum of a non-compact arithmetic hyperbolic orbifold of dimension $n \geqslant 4$ have exponential growth rate $$ \langle g(L) \rangle \geqslant c \frac{e^{([n/2] - 1)L}}{L^{1 + δ_{5, 7}(n) }}, $$ extending the analogous result for even dimensions of Belolipetsky, Lalín, Murillo and Thompson. Our proof is based on the study of (square-rootable) Salem numbers.
As a counterpart, we also prove an asymptotic formula for the distribution of square-rootable Salem numbers by adapting the argument of Götze and Gusakova. It shows that one can not obtain a better estimate on mean multiplicities using our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02568 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multiplicities in the length spectrum and growth rate of Salem numbers Grebennikov, Alexandr Geometric Topology Number Theory 11F06, 11K16, 20H10 We prove that mean multiplicities in the length spectrum of a non-compact arithmetic hyperbolic orbifold of dimension $n \geqslant 4$ have exponential growth rate $$ \langle g(L) \rangle \geqslant c \frac{e^{([n/2] - 1)L}}{L^{1 + δ_{5, 7}(n) }}, $$ extending the analogous result for even dimensions of Belolipetsky, Lalín, Murillo and Thompson. Our proof is based on the study of (square-rootable) Salem numbers. As a counterpart, we also prove an asymptotic formula for the distribution of square-rootable Salem numbers by adapting the argument of Götze and Gusakova. It shows that one can not obtain a better estimate on mean multiplicities using our approach. |
| title | Multiplicities in the length spectrum and growth rate of Salem numbers |
| topic | Geometric Topology Number Theory 11F06, 11K16, 20H10 |
| url | https://arxiv.org/abs/2309.02568 |