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Multiplicities in the Length Spectrum and Growth Rate of Salem Numbers

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Abstract

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

$$\begin{aligned} \langle g(L) \rangle \geqslant c \frac{e^{([n/2] - 1)L}}{L^{1 + \delta _{5, 7}(n) }}, \end{aligned}$$

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.

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Notes

  1. Formally, a closed geodesic is a pair \(([\gamma ], L)\), where \([\gamma ]\) is the equivalence class of \(\gamma \) up to shifts of argument, and L is its period. In other words, a closed geodesic “remembers” its length and orientation, but not its starting point.

  2. An isometry u of \({{\,\mathrm{\mathbb {H}}\,}}^n\) is called an elliptic transformation if its action on \({{\,\mathrm{\mathbb {H}}\,}}^n\) has a fixed point. When u lies in a discrete subgroup of \({{\,\textrm{Isom}\,}}({{\,\mathrm{\mathbb {H}}\,}}^n)\), it is elliptic if and only if it has finite order.

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Acknowledgements

I am grateful to my advisor Mikhail Belolipetsky for bringing my attention to this problem and helpful comments that have significantly improved this note. I am also thankful to the anonymous referees for their careful reading and valuable suggestions. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brasil (CAPES).

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  1. IMPA, Estrada Dona Castorina 110, Jardim Botanico, Rio de Janeiro, Brazil

    Alexandr Grebennikov

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  1. Alexandr Grebennikov
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Correspondence to Alexandr Grebennikov.

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Grebennikov, A. Multiplicities in the Length Spectrum and Growth Rate of Salem Numbers. Bull Braz Math Soc, New Series 55, 25 (2024). https://doi.org/10.1007/s00574-024-00398-4

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  • DOI: https://doi.org/10.1007/s00574-024-00398-4

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