Families of periodic Jacobi–Perron algorithms for all period lengths

For all integers m≥n≥2m≥n≥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 (m1/3,m2/3)(m1/3,m2/3) are also made.