Rational numbers with purely periodic $\beta$-expansion

TL;DR

This paper characterizes cubic Pisot numbers with the property that all sufficiently small positive rationals have purely periodic -expansions, extending previous degree 2 results and analyzing the irrationality of a related supremum.

Contribution

It completes the classification of degree 3 algebraic numbers with purely periodic -expansions for small rationals and investigates the irrationality of the supremum -expansion threshold.

Findings

Characterization of degree 3 algebraic numbers with purely periodic -expansions

Extension of degree 2 results to cubic Pisot units

Proof that (eta) is irrational for certain cubic Pisot units

Abstract

We study real numbers $\beta$ with the curious property that the $\beta$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let $\gamma(\beta)$ denote the supremum of the real numbers $c$ in $(0,1)$ such that all positive rational numbers less than $c$ have a purely periodic $\beta$-expansion. We prove that $\gamma(\beta)$ is irrational for a class of cubic Pisot units that contains the smallest Pisot number $\eta$. This result is motivated by the observation of Akiyama and Scheicher that $\gamma(\eta)=0.666 666 666 086 ...$ is surprisingly close to 2/3.