On the distribution of Salem numbers

In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals $I_1,\ldots, I_{k}\subset \left[0;\pi\right]$,$1\leq k\leq m$let$Sal_{m,k}(Q,I_1,\ldots,I_{k})$ denote the set of ordered $(k+1)$-tuples $({\alpha }_0,\dots ,{\alpha }_k)$ of conjugate algebraic integers, such that ${\alpha }_0$ is a Salem numbers of degree $2m+2$ satisfying $\alpha \le Q$ for some positive real number $Q$ and $\arg {\alpha }_i\in I_i$. We derive the following asymptotic approximation $$\mathrm{\#}Sa{l}_{m,k}(Q,I_1,\dots ,I_k)={\omega }_m{Q}^{m+1}\underset{I_1}{\int }\dots \underset{I_k}{\int }{\rho }_{m,k}(\mathit{\theta })\mathrm{d}\mathit{\theta }+\mathrm{O}\left({\mathrm{Q}}^{\mathrm{m}}\right),\mathrm{Q}\to \mathrm{\infty },$$ providing explicit expressions for the constant ${\omega }_m$ and the function ${\rho }_{m,k}(\mathit{\theta })$. Moreover we derive a similar asymptotic formula for the set of all Salem numbers of fixed degree and absolute value bounded by $Q$ as $Q\to \mathrm{\infty }$.