Multiplicities in the length spectrum and growth rate of Salem numbers

TL;DR

This paper establishes exponential growth rates for mean multiplicities in the length spectrum of certain hyperbolic orbifolds, linking these to properties of Salem numbers, and provides asymptotic distribution results for these numbers.

Contribution

It extends previous results on length spectrum multiplicities to higher dimensions and connects these to Salem numbers, offering new growth rate estimates and distribution asymptotics.

Findings

Mean multiplicities grow exponentially with length in hyperbolic orbifolds.

Established lower bounds for growth rates involving Salem numbers.

Provided asymptotic distribution formulas for square-rootable Salem numbers.

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 $$ \langle g(L) \rangle \geqslant c \frac{e^{([n/2] - 1)L}}{L^{1 + \delta_{5, 7}(n) }}, $$ 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.