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Multidimensional continued fractions. (English) Zbl 0981.11029

Oxford Science Publications. Oxford: Oxford University Press. viii, 234 p. (2000).
A multidimensional generalization of the continued fraction algorithm was first studied by C. G. J. Jacobi in the 19th century. Jacobi’s work, which was published posthumously, was aimed at generalizing Lagrange’s theorem on periodic continued fraction expansions. Jacobi’s approach was taken up by O. Perron in his seminal 1907 paper, leading to the Jacobi-Perron algorithm (JPA). Apart from the JPA there are numerous other multidimensional generalizations of the continued fraction algorithm. A good survey (up to 1981) of these algorithms can be found in A. J. Brentjes’ monograph [Multidimensional continued fraction algorithms (1981; Zbl 0471.10024)].
The present book is to some extent a continuation of F. Schweiger’s earlier book [Ergodic theory of fibred systems and metric number theory (1995; Zbl 0819.11027)], in which the ergodic theory underlying number theoretic expansions is discussed in great detail. The present book – which only has a slight overlap with Brentjes’ monograph (see Chapter 2), and which can be read independently from both Brentjes’ book as well as Schweiger’s earlier book – deals with multidimensional generalizations of the continued fraction algorithm that can be described by fractional linear maps (or equivalently: by a set of \((n+1)\times (n+1)\) matrices that generalize the matrices \(\left(\begin{smallmatrix} 0&1\\ 1&a \end{smallmatrix} \right)\) associated with the continued fraction algorithm). This gives a general framework in which many of the important generalizations of the continued fraction algorithm can be studied in both their algebraic and algorithmic aspects, as well as their underlying ergodic systems: the JPA (Chapter 4), the algorithms of Güting (Chapter 5), Brun (Chapter 6), Selmer (Chapter 7), the generalized and fully subtractive algorithms (Chapters 8 and 9), and Poincaré’s algorithm (Chapter 21). The book deals with the difficult problems of periodicity and convergence of these algorithms. It is shown that problems of convergence can be redefined to the question whether these algorithms give ‘good’ simultaneous Diophantine approximation.
This book contains a wealth of material on multidimensional continued fraction algorithms and will serve as a reference book for anyone interested in this subject in the years to come.

MSC:

11K50 Metric theory of continued fractions
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11J70 Continued fractions and generalizations
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