Beta-expansions of rational numbers in quadratic Pisot bases

TL;DR

This paper characterizes when rational numbers have purely periodic beta-expansions in quadratic Pisot bases, providing a necessary and sufficient condition and an algorithm for computing this property.

Contribution

It offers a complete characterization of purely periodic expansions for rational numbers in quadratic Pisot bases with specific algebraic properties and introduces an algorithm for computing this.

Findings

Necessary and sufficient condition for pure periodicity when b3(b2)=1

Algorithm to determine b3(b2) for quadratic Pisot bases

Identification of bases where all rationals with certain denominators have pure periodic expansions

Abstract

We study rational numbers with purely periodic R\'enyi $\beta$-expansions. For bases $\beta$ satisfying $\beta^2=a\beta+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $\gamma(\beta)=1$, i.e., that all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a purely periodic $\beta$-expansion. A simple algorithm for determining the value of $\gamma(\beta)$ for all quadratic Pisot numbers $\beta$ is described.