Finiteness and periodicity of continued fractions over quadratic number fields

Abstract: We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$; a particular example of these continued fractions is the $\beta$-continued fraction introduced by Bernat. 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. We also show that for certain four 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 a joint work with Zuzana Masáková and Tomáš Vávra.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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