Periodic expansion of one by Salem numbers

TL;DR

This paper proves that powers of Salem numbers become Parry numbers with positive density, providing bounds for degree 6 and exploring connections to discretized rotations in four dimensions.

Contribution

It establishes that for Salem numbers, a positive density of their powers are Parry numbers, and offers specific bounds and geometric relations.

Findings

For Salem numbers of degree d, powers are Parry numbers with density ≥ c(d).

For degree 6, the density c(6) exceeds 1/2.

Connections between Salem numbers and discretized rotations in 4D are discussed.

Abstract

We show that for a Salem number $\beta$ of degree $d$, there exists a positive constant $c(d)$ that $\beta^m$ is a Parry number for integers $m$ of natural density $\ge c(d)$. Further, we show $c(6)>1/2$ and discuss a relation to the discretized rotation in dimension $4$.