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

Autor: Cornelia Mihaila, Thomas Garrity, Matthew Stoffregen, Nicholas Neumann-Chun, Krishna Dasaratha, Sarah Peluse, Chansoo Lee, Laure Flapan
Rok vydání: 2014
Předmět:
Zdroj: Monatshefte für Mathematik. 174:549-566
ISSN: 1436-5081
0026-9255
DOI: 10.1007/s00605-014-0643-1
Popis: 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 a so that either (a, a^2) or (a, a-a^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. Thus 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.
Comment: 14 pages; New section on earlier work added; to appear in Monatshefte f\"ur Mathematik
Databáze: OpenAIRE
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