Approximation of irrationals (Q1072587)

In her Ph.D. thesis [see also Pac. J. Math. 110, 21-40 (1984; Zbl 0488.12003)], the author has modified the Jacobi-Perron algorithm (JPA) in order to work within the field of complex numbers, and calls this algorithm a generalized Euclidean algorithm (GEA). As for JPA, it is possible to evaluate the limit of the ''convergents'' of GEA. In fact, the author considers the fields \({\mathbb{Q}}(\omega)\) where \(\omega =(D^ n+1)^{1/n}\), \(D\in {\mathbb{N}}\) and \(n\in \{2,3,5\}\), and develops GEA of a judiciously chosen starting vector; the development is purely periodic of length 1, which is no surprise since it is a JPA result which can be found for arbitrary n in \textit{L. Bernstein}'s ''The Jacobi-Perron algorithm; its theory and application'' (Lect. Notes Math. 207) (1971; Zbl 0213.052). Then she writes in \textit{explicit} form the so-called ''matricians'' which are known to satisfy a linear recurrence of order n, from which she can deduce explicit approximations of certain irrationals. The calculations involved are elementary and would have been interesting, had they been performed for arbitrary n.