Alphabets, rewriting trails and periodic representations in algebraic bases

TL;DR

This paper investigates the structure of alphabets and representations in algebraic bases, introducing rewriting trails and analyzing their relation to algebraic properties, with applications to Galois conjugation and dynamical zeta functions.

Contribution

It introduces the concept of rewriting trails for constructing intermediate alphabets and explores their implications for algebraic base representations and related number theoretic problems.

Findings

Maximal alphabets relate to Pierce numbers and Lehmer's problem.

Rewriting trails help construct intermediate alphabets with small polynomial values.

Poles of the dynamical zeta function are roots of the minimal polynomial of the base.

Abstract

For $\beta > 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(\beta) = {\rm Per}_{\mathcal{A}}(\beta)$, where ${\rm Per}_{\mathcal{A}}(\beta)$ is the set of complex numbers which are $(\beta, \mathcal{A})$-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and maximal alphabets are defined. The maximal alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base $\beta$ and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in $\mathbb{Q}(\beta)$, generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases $\gamma_s := \gamma_{n, m_1 , \ldots , m_s}$ such that $\gamma_{s}^{-1}$ is the unique root in $(0,1)$ of an almost Newman polynomial of the type $-1+x+x^n +x^{m_1}+\ldots+ x^{m_s}$, $n \geq 3$, $s \geq 1$, $m_1 - n \geq n-1$, $m_{q+1}-m_q \geq n-1$ for all $q \geq 1$. For $\beta > 1$ a reciprocal algebraic integer close to one, the poles of modulus $< 1$ of the dynamical zeta function of the $\beta$-shift $\zeta_{\beta}(z)$ are shown, under some assumptions, to be zeroes of the minimal polynomial of $\beta$.