Finiteness and periodicity of continued fractions over quadratic number fields

TL;DR

This paper generalizes classical results on continued fractions to quadratic number fields, proving finiteness and periodicity properties for $eta$-continued fractions and answering longstanding questions about their representations.

Contribution

It extends the classical Lagrange theorem to quadratic fields and characterizes finiteness and periodicity of $eta$-continued fractions for quadratic Perron numbers.

Findings

$eta$-continued fractions are finite or eventually periodic for quadratic Perron numbers

Certain quadratic Perron numbers have all elements of $ ext{Q}(eta)$ finitely represented

Answers to questions of Rosen and Bernat regarding continued fraction representations

Abstract

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $\beta$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $\beta$, the $\beta$-continued fraction expansion of elements in $\mathbb Q(\beta)$ is either finite of eventually periodic. The same holds for $\beta$ being a square root of an integer. We also show that for certain 4 quadratic Perron numbers $\beta$, the $\beta$-continued fraction represents finitely all elements of the quadratic field $\mathbb Q(\beta)$, thus answering questions of Rosen and Bernat. Based on the validity of a conjecture of Mercat, these are all quadratic Perron numbers with this feature.