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.
Beyond the Erdős–Sós conjecture
Authors
Davoodi, A.; Piguet, D.; Řada, H.; Sanhueza-Matamala, N.
Year
2023
Published
Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications. Brno: Masarykova univerzita, 2023. p. 328-335. ISSN 2788-3116. ISBN 978-80-280-0344-9.
Type
Proceedings paper
Departments
Annotation
We prove an asymptotic version of a tree-containment conjecture of Klimošová, Piguet and Rozhoň [European J. Combin. 88 (2020), 103106] for graphs with quadratically many edges. The result implies that the asymptotic version of the Erdős-Sós conjecture in the setting of dense graphs is correct.
Permutation flip processes
Authors
Hladký, J.; Řada, H.
Year
2023
Published
Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications. Brno: Masarykova univerzita, 2023. p. 587-594. ISSN 2788-3116. ISBN 978-80-280-0344-9.
Type
Proceedings paper
Departments
Annotation
We introduce a broad class of stochastic processes on permutations which we call flip processes. A single step in these processes is given by a local change on a randomly chosen fixed-sized tuple of the domain. We use the theory of permutons to describe the typical evolution of any such flip process started from any initial permutation.
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.