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 $I_1,\dots ,I_k\subset \left[0;\pi \right]$, $1\le k\le m$ let $Sa{l}_{m,k}(Q,I_1,\dots ,I_k)$ denote the set of ordered $(k+1)$-tuples $\left({\alpha }_0,\dots ,{\alpha }_k\right)$ 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 \[ \# Sal_{m,k}(Q,I_1,\ldots,I_{k})=ω_m\,Q^{m+1}\,\int\limits_{I_1}\ldots\int\limits_{I_{k}}ρ_{m,k}(\boldsymbolθ)\rm d\boldsymbolθ+O\left(Q^{m}\right),\quad Q\rightarrow\infty, \] providing explicit expressions for the constant ${\omega }_m$ and the function ${\rho }_{m,k}(\mathbf{\theta })$. Moreover we derive a similar asymptotic formula for the set of all Salem numbers of fixed degree and absolute value bounded by $Q$ as $Q\to \infty$.