Approximation by algebraic numbers.

*(English)*Zbl 1055.11002
Cambridge Tracts in Mathematics 160. Cambridge: Cambridge University Press (ISBN 0-521-82329-3/hbk). xv, 274 p. (2004).

The main focus of this book is the approximation of real numbers by rational or algebraic numbers. There is also a short section on the approximation of complex numbers, of \(p\)-adic numbers and of formal power series. Part of the subject is classical, going back to the early stages of the theory of continued fractions, with a number of important recent developments, most of which are covered in this book. A number of exercises are included, they provide an introduction to recent research works. There are many open problems. The list of more than 600 references shows the vitality of the subject.

In Chapter 1 the theorems of Dirichlet, Liouville, Khinchine and the Duffin-Schaeffer conjecture are discussed. Next in Chapter 2 the author states the main results on the approximation to algebraic numbers, including results by Thue, Siegel, Roth, Ridout, Feldman, Wirsing, Schmidt, Baker, Bombieri and others. Chapter 3 is devoted to Mahler’s classification of transcendental numbers, including a comparison with Koksma’s classification and a discussion of exponents of Diophantine approximation. Chapter 4 deals with Mahler’s conjecture on \(S\)-numbers and the works of Sprindzhuk, Bernik, Kleinbock and Margulis. In Chapter 5 the author considers the Hausdorff dimension in order to be able to distinguish between different sets of Lebesgue measure zero. The JarnĂk-Besicovitch Theorem is proved. Deeper results on the measure of exceptional sets are provided in Chapter 6, while Chapter 7 is devoted to \(T\)- and \(U\)-numbers. Other classification of transcendental numbers, due to Sprindzhuk, Mahler, Philippon are introduced in Chapter 8. After Chapter 9 which deals with approximation in other fields, Chapter 10 includes a number of conjectures and open questions with no less than 54 problems. There is an appendix on the geometry of numbers.

Part of the material covered in this book is basic, but the author revisited it and included many personal contributions which make this book useful not only for those who wish to learn the subject, but also for the specialists of Diophantine approximation.

In Chapter 1 the theorems of Dirichlet, Liouville, Khinchine and the Duffin-Schaeffer conjecture are discussed. Next in Chapter 2 the author states the main results on the approximation to algebraic numbers, including results by Thue, Siegel, Roth, Ridout, Feldman, Wirsing, Schmidt, Baker, Bombieri and others. Chapter 3 is devoted to Mahler’s classification of transcendental numbers, including a comparison with Koksma’s classification and a discussion of exponents of Diophantine approximation. Chapter 4 deals with Mahler’s conjecture on \(S\)-numbers and the works of Sprindzhuk, Bernik, Kleinbock and Margulis. In Chapter 5 the author considers the Hausdorff dimension in order to be able to distinguish between different sets of Lebesgue measure zero. The JarnĂk-Besicovitch Theorem is proved. Deeper results on the measure of exceptional sets are provided in Chapter 6, while Chapter 7 is devoted to \(T\)- and \(U\)-numbers. Other classification of transcendental numbers, due to Sprindzhuk, Mahler, Philippon are introduced in Chapter 8. After Chapter 9 which deals with approximation in other fields, Chapter 10 includes a number of conjectures and open questions with no less than 54 problems. There is an appendix on the geometry of numbers.

Part of the material covered in this book is basic, but the author revisited it and included many personal contributions which make this book useful not only for those who wish to learn the subject, but also for the specialists of Diophantine approximation.

Reviewer: Michel Waldschmidt (Paris)

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11J04 | Homogeneous approximation to one number |

11J13 | Simultaneous homogeneous approximation, linear forms |

11J17 | Approximation by numbers from a fixed field |

11J25 | Diophantine inequalities |

11J68 | Approximation to algebraic numbers |

11J83 | Metric theory |

11K60 | Diophantine approximation in probabilistic number theory |