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 , . . . , I k ⊂ [0; π], 1 ≤ k ≤ m let Sal m,k (Q, I 1 , . . . , I k ) denote the set of ordered (k + 1)-tuples (α 0 , . . . , α k ) of conjugate algebraic integers, such that α 0 is a Salem numbers of degree 2m + 2 satisfying α ≤ Q for some positive real number Q and arg α i ∈ I i . We derive the following asymptotic approximationproviding explicit expressions for the constant ω m and the function ρ m,k (θ). 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 → ∞.

Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds

“…Now by Emery et al ( 2019 , Theorem 1.1) each geodesic corresponds to a Salem number of degree (here we use that n is even and that the degrees of the Salem numbers are even). By Götze and Gusakova ( 2019 , Theorem 1.1) the total number of such Salem numbers is bounded by . Hence, on average, the geodesic lengths have to appear with multiplicity at least .…”

“…His experiments led to a set of interesting problems and conjectures, some of which were successfully resolved by Calegari and Huang ( 2017 ). Later on, some ideas from their approach helped Götze and Gusakova ( 2019 ) to compute the asymptotic growth of Salem numbers. The precise form of their result is given in Theorem 4 .…”

Counting Salem numbers of arithmetic hyperbolic 3-orbifolds

“…His experiments led to a set of interesting problems and conjectures, some of which were successfully resolved by F. Calegari and Z. Huang in [CH17]. Later on, some ideas from their approach helped F. Götze and A. Gusakova to compute the asymptotic growth of Salem numbers in [GG19]. The precise form of their result is given in Theorem 2.…”