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

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.