A Non-Trivial Minoration for the Set of Salem Numbers

Abstract

The set of Salem numbers is proved to be bounded from below by ${\theta }_{31}^{-1}=1.08544\dots$ where ${\theta }_n$, $n\ge 2$, is the unique root in $(0,1)$ of the trinomial $-1+x+x^n$. Lehmer's number $1.176280\dots$ belongs to the interval $({\theta }_{12}^{-1},{\theta }_{11}^{-1})$. We conjecture that there is no Salem number in $({\theta }_{31}^{-1},{\theta }_{12}^{-1})=(1.08544\dots ,1.17295\dots )$. 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.

arXiv commenttext dedicated to the 75th birthday of Christiane Frougny - NUMERATION2023 -. arXiv admin note: text overlap with arXiv:1911.10590