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

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 α so that either (α, α 2 ) or (α, α − α 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.

Functional analysis behind a family of multidimensional continued fractions. Part I

“…As this is background, much of this section is similar to certain sections in [3], [6], [12]- [13], [16], [25] and [39].…”

“…Triangle partition maps ( [12]- [13], [25]) are a family of 216 multidimensional continued fraction algorithms that include (when combinations are allowed) many, if not most, well-known multidimensional continued fraction algorithms, which is why they are a natural class of algorithms to study. These maps are reviewed in Section 2.…”

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

“…The above development of the triangle sequence is called the additive approach. In [9,5,6,7,14], the multiplicative version is used. In the multiplicative version, we look at the subtriangles…”

On the periodic writing of cubic irrationals and a generalization of Rédei functions

“…See the beautiful book of Schweiger [31] for a guide about multidimensional continued fractions and [21] for a new geometric vision of multidimensional continued fractions. A very interesting approach can be found in the works [15], [3], [13], where a multidimensional continued fraction related to triangle sequences is studied. Moreover, in [6] a generalization of the Minkowski question-mark function is developed.…”