A Non-Trivial Minoration for the Set of Salem Numbers

2024/01/11 by Jean-Louis Verger-Gaugry, Verger-Gaugry, Jean-Louis
Computer Science · Mathematics · #03D45 #11K16 #11M41 #11R06 #11R09 #30B10 #30B40 #37C30 #37N99 #Advanced Algebra and Logic #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2401.05843

openalex publication_date 2024/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The set of Salem numbers is proved to be bounded from below by θ₃₁^-1= 1.08544… where θ_n, n ≥ 2, is the unique root in (0,1) of the trinomial -1+x+xⁿ. Lehmer's number 1.176280… belongs to the interval (θ₁₂^-1, θ₁₁^-1). We conjecture that there is no Salem number in (θ₃₁^-1, θ₁₂^-1) = (1.08544…, 1.17295…). For proving the Main Theorem, the algebraic and analytic properties of the dynamical zeta function of the Rényi-Parry numeration system are used, with real bases running over the set of real reciprocal algebraic integers, and variable tending to 1.

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