Families of periodic Jacobi-Perron algorithms for all period lengths

TL;DR

This paper constructs infinite families of purely periodic Jacobi-Perron Algorithm expansions for all period lengths, revealing new units and periodicity properties in higher dimensions.

Contribution

It introduces explicit families of periodic JPA expansions for all period lengths and explores related algebraic units and periodicity phenomena.

Findings

Infinite families of periodic JPA expansions for all period lengths

Identification of associated Hasse-Bernstein units

Observations on units of Levesque-Rhin and periodicity of specific expansions

Abstract

For all integers $m \geq n \geq 2$, we exhibit infinite families of purely periodic Jacobi-Perron Algorithm (JPA) expansions of dimension $n$ with period length equal to $m$ along with the associated Hasse-Bernstein units. Some observations on the units of Levesque-Rhin as well as the periodicity of the JPA expansion of $\left( m^{1/3}, m^{2/3} \right)$ are also made.