On the co-factors of degree 6 Salem number beta expansions

TL;DR

This paper explores the properties of degree 6 Salem numbers' beta expansions, providing computational methods, theoretical results on period lengths, and evidence of very long yet eventually periodic expansions.

Contribution

It introduces computational techniques for identifying degree 6 Salem numbers with periodic expansions and proves that their expansions can have arbitrarily large periods.

Findings

Degree 6 Salem numbers can have arbitrarily large periods in their expansions.

Examples with period and preperiod lengths exceeding 10^{10} are found.

Computational methods help identify families of such Salem numbers.

Abstract

For $\beta > 1$, a sequence $(c_n)_{n \geq 1} \in \mathbb{Z}^{\mathbb{N}^+}$ with $0 \leq c_n < \beta$ is the \emph{beta expansion} of $x$ with respect to $\beta$ if $x = \sum_{n = 1}^\infty c_n\beta^{-n}$. Defining $d_\beta(x)$ to be the greedy beta expansion of $x$ with respect to $\beta$, it is known that $d_\beta(1)$ is eventually periodic as long as $\beta$ is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proportion of degree 8 Salem numbers do not. In this paper, we investigate the degree 6 case. We present computational methods for searching for families of degree 6 numbers with eventually periodic greedy expansions by studying the co-factors of their expansions. We also prove that the greedy expansions of degree 6 Salem numbers can have arbitrarily large periods. In addition, computational evidence is compiled on the set of degree 6 Salem numbers with $\text{trace}(\beta) \leq 15$. We give examples of numbers with $\text{trace}(\beta) \leq 15$ whose expansions have period and preperiod lengths exceeding $10^{10}$, yet are still eventually periodic.