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Continued fractions. (English) Zbl 1161.11028

Hackensack, NJ: World Scientific (ISBN 981-256-477-2/hbk). xiii, 245 p. (2006).
This book strives to put together new results on continued fractions, their algorithms, the distribution of their approximants, approximation results etc., that were found over the last decade and have been published scattered through the literature. The book is concerned with the algebraic aspects of the algorithms (and ergodicity of attached operators etc.) and not with the analytic aspects. The way things fit together makes it a nicely written book that should be available to every researcher in the field. Below the chapters will be described separately.
1. Introduction (21 pages): Treats the continued fraction algorithm for a real number and the special case of a quadratic irrational number, connection with the Pell equation, linear recurrence relations and diophantine approximation (badly approximable, discrepancy, etc.)
2. Generalizations of the gcd and the Euclidean algorithm (10 pages): Gauss rational approximation for a complex number (using the Gaussian integers as entries), gcd of a pair of algebraic integers (in the setting of an algebraic number field, a finite extension over \(\mathbb{Q}\)); lattice reduction algorithms are mentioned for the first time.
3. Continued fractions with small partial quotients (16 pages): The continued fractions looked into can be integer based, fractions with numerators and denominators polynomials from a finite field or formal power series; the notion ‘small’ varies from case to case. The definition of discrepancy \(\text{D}_N^{\ast}\) for a sequence \((x_k)\) in \([0,1)\) in the one-dimensional case used here, is \[ \text{D}_N^{\ast}(x)=\max_{0\leq t\leq 1}\,\left| N^{-1}\#\{k\,:\,1\leq k\leq N\text{ and }0\leq x_k<t\}-t\right| . \]
4. Ergodic Theory (17 pages): The information necessary for the theory used in the book is given (dynamical systems, ergodicity, (weak-) mixing, etc.); application to the distribution of \(\theta_n(x)=q_n| p_n-x q_n| \) (\(p_n,\,q_n\) numerator and denominator of the \(n\)th continued fraction convergent to \(x\in E_M,\;E_M\) consists of all numbers in \([0,1]\) with infinite continued fraction for which all partial quotients come from a fixed set \(M\) with special properties. A proof of a generalization (due to the author, 1998) of a theorem by Nair (1997) is given: for any strictly increasing sequence \((n_j)\) one has \[ \lim_{N\rightarrow\infty}\,{1\over N}\,\#\{1\leq j\leq N\,:\,\theta_{n_j}(x)\leq z\}=f(z), \] with \[ f(z)=\begin{cases} z/\log{2} & 0\leq z\leq 1/2 \cr (1-z+\log{(2z)})/\log{2} & 1/2\leq z\leq 1 \cr 1 & z>1\end{cases}. \] (Bosman, Jager and Wiedijk (1983) proved this for \(\theta_n(x)\) and almost all \(x\) from \([0,1]\))
5. Complex continued fractions (28 pages): Main use of the Schmidt regular chains algorithm (resembling the Lagarias multidimensional continued fraction algorithm) and the Hurwitz complex continued fraction. The distribution of the remainders is investigated and a class of algebraic approximants of an atypical Hurwitz continued fraction is studied. Finally the Gauss-Kuzmin density for the Hurwitz algorithm is derived.
6. Multidimensional Diophantine approximation (28 pages): In higher dimensions an important role is played by lattice reduction algorithms (Lenstra-Lenstra-Lovasz procedure connected with the Gram-Schmidt algorithm, Minkowski reduced bases connected with Gauss lattice reduction and the Lagarias procedure). The different approaches are studied in detail and in connection with simultaneous Diophantine approximation the names of J. W. S. Cassels and W. M. Schmidt appear (the Szekeres algorithm is not mentioned, the Jacobi-Perron algorithm only occasionally).
7. Powers of an algebraic integer (17 pages) This is concerned with so called good triples. For \(\alpha\in\mathbb{R}\), algebraic of degree \(n\), the triple \((\sigma,q,\alpha),\;\sigma>0,\,q>0\) an integer, is called good if there exist integers \(p_1,p_2,\ldots\) such that \[ \left| p_j-q\alpha^j\right| \leq \sigma q^{-1/(n-1)},\;1\leq j\leq n-1. \] (the pigeon-hole principle implies; for any \(\sigma\geq 1\) there is an infinite sequence of positive integers \(q\) with \((\sigma,q,\alpha)\) good). Also good units and good denominators are studied and it is proved that the ratio of two consecutive good \(q\)’s is asymptotically close to an element of a finite subset of \(\mathbb{Q}(\alpha)\).
8. Marshall Hall’s theorem (10 pages): Deals with statements like \(F_4+F_4=\mathbb{R}\) (Hall, 1947) where \(F_N=E_N+\mathbb{Z}\) and \(E_N\) the continued fraction Cantor set \[ E_N=\{[a_1,\ldots,a_n]\,| \,1\leq a_k\leq N\text{ for }k\geq 1\}. \]
9. Function-analytic techniques (50 pages): This covers continued fraction Cantor sets and their Hausdorff and Minkowski dimensions, Hilbert spaces of power series, positive operators, the unform spectral gap and the convexity of the spectral radius \(\lambda_M\).
10. The generating function method (7 pages): Connection with entropy of a dynamical system and results on resolvents \((I-\omega H_s)^{-1}\) and asymptotics for partial sums of certain Dirichlet series at \(s=1\).
11. Conformal iterated function systems (4 pages): The concept of ‘porous’ and a result on the doubling property of a measure (for the measure \(\nu\) on \(X\) there exists a \(c>0\) such that for almost-\(\nu\)-all \(x\in X\) and all \(0 < \varepsilon < 1\) we have \(\nu([x-2\varepsilon,x+2\varepsilon])\leq c\nu([x-\varepsilon,x+\varepsilon])\)).
12. Convergence of continued fractions (15 pages): Some classical results are stated and the reader is referred to standard methods that can be found in the classical references (Perron, part 1; Jones and Thron).
References (7 pages): There are 135 references.

MSC:

11K50 Metric theory of continued fractions
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11K60 Diophantine approximation in probabilistic number theory
11J70 Continued fractions and generalizations
28D05 Measure-preserving transformations
30B70 Continued fractions; complex-analytic aspects
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
40A15 Convergence and divergence of continued fractions
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