Cubic Irrationals and Periodicity via a Family of Multi-dimensional Continued Fraction Algorithms

TL;DR

This paper introduces a family of multi-dimensional continued fraction algorithms that identify and analyze cubic irrational numbers with periodic expansions, linking algebraic number theory with algorithmic continued fractions.

Contribution

It constructs a new family of algorithms that connect cubic irrationals with periodic continued fractions and units in cubic number fields, expanding understanding of their structure.

Findings

Identified classes of cubic irrationals with periodic continued fraction expansions.

Established a link between periodicity and units in cubic number fields.

Recast the problem as matrix periodicity with nonnegative integer entries.

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 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.