Periodicity of general multidimensional continued fractions using repetend matrix form
Authors
Řada, H.; Starosta, Š.; Kala, V.
Year
2024
Published
EXPOSITIONES MATHEMATICAE. 2024, 42(3), ISSN 0723-0869.
Type
Article
Departments
Annotation
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend and use it to prove that a number of vectors have an eventually periodic expansion in the Algebraic Jacobi–Perron algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive.
Matrix Form of Multidimensional Continued Fractions
Authors
Year
2022
Published
Doktorandské dny 2022. Praha: CTU. Faculty of Nuclear Sciences and Physical Engineering, 2022. p. 113-124.
Type
Proceedings paper
Departments
Bounds on the period of the continued fraction after a Möbius transformation
Authors
Year
2020
Published
Journal of Number Theory. 2020, 212 122-172. ISSN 0022-314X.
Type
Article
Departments
Annotation
We study Möbius transformations (also known as linear fractional transformations) of quadratic numbers. We construct explicit upper and lower bounds on the period of the continued fraction expansion of a transformed number as a function of the period of the continued fraction expansion of the original number. We provide examples that show that the bound is sharp.