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The Jacobi–Perron algorithm and Pisot numbers. (English) Zbl 1051.11037

The Jacobi-Perron algorithm 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.
In this paper, the authors prove that for \(n= 2\) this unit is a Pisot number (a positive algebraic integer with each conjugate in \(| z|< 1\)) and that it is not necessary the case for \(n\geq 3\).
Recently, B. Adam and G. Rhin found a method yielding all pairs of real numbers with periodic Jacobi-Perron algorithm which produce a given unit in a real cubic field. In many examples they get no set with periodic Jacobi-Perron algorithm when the given unit is not a Pisot number. So, they ask if this is always true. By this paper, the authors give a positive answer to their question for \(n= 2\) and they prove that this is not always the case for \(n\geq 3\).

MSC:

11J70 Continued fractions and generalizations
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R27 Units and factorization
11R16 Cubic and quartic extensions
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