On the distribution of Salem numbers

Abstract

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 $\left(\alpha_0,\ldots,\alpha_{k}\right)$ of conjugate algebraic integers, such that $\alpha_0$ is a Salem numbers of degree $2m+2$ satisfying $\alpha\leq Q$ for some positive real number $Q$ and $\arg\alpha_i\in I_i$. We derive the following asymptotic approximation \[ \# Sal_{m,k}(Q,I_1,\ldots,I_{k})=\omega_m\,Q^{m+1}\,\int\limits_{I_1}\ldots\int\limits_{I_{k}}\rho_{m,k}(\boldsymbol\theta)\rm d\boldsymbol\theta+O\left(Q^{m}\right),\quad Q\rightarrow\infty, \] providing explicit expressions for the constant $\omega_m$ and the function $\rho_{m,k}(\boldsymbol\theta)$. Moreover we derive a similar asymptotic formula for the set of all Salem numbers of fixed degree and absolute value bounded by $Q$ as $Q\rightarrow\infty$.