Multiplicities in the length spectrum and growth rate of Salem numbers
| Autor: | Grebennikov, Alexandr |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Bulletin of the Brazilian Mathematical Society, New Series, volume 55, article number 25 (2024) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00574-024-00398-4 |
| Popis: | 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 $$ \langle g(L) \rangle \geqslant c \frac{e^{([n/2] - 1)L}}{L^{1 + \delta_{5, 7}(n) }}, $$ extending the analogous result for even dimensions of Belolipetsky, Lal\'in, 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 G\"otze and Gusakova. It shows that one can not obtain a better estimate on mean multiplicities using our approach. Comment: v3: 20 pages, final version; more detailed, used as Master's thesis. v2: 18 pages, journal version; introduction expanded, statements of the main results slightly modified, other minor corrections |
| Databáze: | arXiv |
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