Multiplicities in the length spectrum and growth rate of Salem numbers

We prove that mean multiplicities in the length spectrum of a non-compact arithmetic hyperbolic orbifold of dimension $n⩾4$ have exponential growth rate $$⟨g(L)⟩⩾c\frac{e^{([n\mathrm{/}2]-1)L}}{L^{1+{\delta }_{5,7}(n)}},$$ 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.