2019/04/05 by Friedrich Götze, Anna Gusakova
Mathematics · #Advanced Combinatorial Mathematics #Algebraic number #Asymptotic formula #Bounded function #Combinatorics #Constant (computer programming) #Degree (music) #Discrete mathematics #Disjoint sets #Distribution (mathematics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Stochastic processes and statistical mechanics #Tuple #math.NT
paper · pdf · doi:10.1016/j.jnt.2020.02.012
published as Journal of Number Theory, vol. 216, pp. 192 - 215 (2020)
arxiv created 2019/04/05 · openalex publication_date 2020/03/17 · arxiv updated 2021/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals I1,…, Ik⊂ [0;π], 1≤ k≤ m let Salm,k(Q,I1,…,Ik) denote the set of ordered (k+1)-tuples (α0,…,αk) of conjugate algebraic integers, such that α0 is a Salem numbers of degree 2m+2 satisfying α≤ Q for some positive real number Q and argαi∈ Ii. We derive the following asymptotic approximation # Salm,k(Q,I1,…,Ik)=ωm Qm+1 ∫I1…∫_Ikρm,k(\boldsymbolθ)\rm d\boldsymbolθ+O(Qm), Q→∞, providing explicit expressions for the constant ωm and the function ρm,k(\boldsymbolθ). 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→∞.