In this paper, we prove a periodicity theorem for certain continued fractions with partial quotients in the ring of integers of a fixed quadratic field. This theorem generalizes the classical theorem of Lagrange to a large set of continued fraction expansions.As an application we consider the beta-continued fractions and show that for any quadratic Perron number beta, the beta-continued fraction expansion of elements in Q(beta) is either finite or eventually periodic.
FINITENESS AND PERIODICITY OF CONTINUED FRACTIONS OVER QUADRATIC NUMBER FIELDS
Veneziano F.
2022-01-01
Abstract
In this paper, we prove a periodicity theorem for certain continued fractions with partial quotients in the ring of integers of a fixed quadratic field. This theorem generalizes the classical theorem of Lagrange to a large set of continued fraction expansions.As an application we consider the beta-continued fractions and show that for any quadratic Perron number beta, the beta-continued fraction expansion of elements in Q(beta) is either finite or eventually periodic.
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