The Jacobi–Perron Algorithm and Pisot numbers
Eugène Dubois, Ahmed Farhane, Roger Paysant-Le Roux · Acta Arithmetica · 2004
The Jacobi-Perron Algorithm introduced by Jacobi [7] and O. Perron [9] is a generalization of the continued fraction algorithm.Applied to an n-uple of real numbers, it gives simultaneous approximations.In case of periodicity it yields a unit of a number field which commands the quality of simultaneous approximations.We prove that for n = 2 this unit is a Pisot number (positive algebraic integer with each conjugate in |z| < 1) and that this is not necessarily the case for n ≥ 3.The problem of characterizing the periodicity of JPA (Jacobi-Perron Algorithm) is still open for n ≥ 2. Many families of sets of n real numbers for which the JPA is periodic were found by L. Bernstein [2], E. Dubois & R. Paysant-Le Roux [5], C. Levesque & G. Rhin [8].M. Bouhamza for n = 3 and n = 4 [3, 4], and then E. Dubois and R. Paysant-Le Roux for every n [6] proved that there exists, in any real number field of degree n + 1, an n-uple of real numbers with periodic JPA.Recently B. Adam & G. Rhin [1] found a method yielding all pairs of real numbers with periodic JPA which produce a given unit in a real cubic field.In many examples they get no set with periodic JPA when the given unit is not a Pisot number.So they ask if this is always true.In this paper we give a positive answer in case n = 2 and we prove that this is not always true for n ≥ 3.I. The Jacobi-Perron Algorithm.The continued fraction algorithm, applied to an irrational real number α, yields a sequence (α k ) k≥0 of real numbers, a sequence (a k ) k≥0 of integers and a sequence (p k /q k ) k≥-2 of