NDLI: Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms

Content Provider	Springer Nature Link
Author	Mihaila, Cornelia Peluse, Sarah Flapan, Laure Neumann Chun, Nicholas Garrity, Thomas Stoffregen, Matthew Dasaratha, Krishna Lee, Chansoo
Copyright Year	2014
Abstract	We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers $$\alpha $$ so that either $$(\alpha , \alpha ^2)$$ or $$(\alpha , \alpha -\alpha ^2)$$ is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair $$(u, u')$$ with $$u$$ a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that $$(u, u')$$ has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. These results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.
Ending Page	566
Page Count	18
Starting Page	549
File Format	PDF
ISSN	00269255
e-ISSN	14365081
Journal	Monatshefte für Mathematik
Issue Number	4
Volume Number	174
Language	English
Publisher	Springer Vienna
Publisher Date	2014-05-31
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cubic number fields Continued fractions and generalizations Hermite problem Multidimensional continued fractions Mathematics Continued fractions
Content Type	Text
Resource Type	Article
Subject	Mathematics

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