Introduction to diophantine approximations.
New exp. ed.

*(English)*Zbl 0826.11030
New York: Springer Verlag. 130 p. (1995).

[For a review of the first ed. (1966) see Zbl 0144.040.]

As pointed out in the foreword to this new second edition, the only change from the first is “the addition of two papers, written in collaboration with W. Adams and H. Trotter, giving computational information for the behavior of certain algebraic and classical transcendental numbers with respect to approximation by rational numbers and their continued fraction”. These papers are [W. Adams and S. Lang, Some computations in diophantine approximations, J. Reine Angew. Math. 220, 163-173 (1965; Zbl 0142.295); S. Lang and H. Trotter, J. Reine Angew. Math. 255, 112-134 (1972; Zbl 0237.10022); Addendum, ibid. 267, 219-220 (1974; Zbl 0305.10026)].

Contents: Chapter 1: General formalism (Rational continued fractions, The continued fraction of a real number, Equivalent numbers, Intermediate convergents).

Chapter II: Asymptotic approximations (Distribution of the convergents, Numbers of constant type, Asymptotic approximations, Relation with continued fractions).

Chapter III: Estimates of averaging sums (The sum of the remainders, The sum of the reciprocals, Quadratic exponential sums, Sums with more general functions).

Chapter IV: Quadratic irrationalities (Quadratic numbers of periodicity, Units and continued fractions, The basic asymptotic estimate).

Chapter V: The exponential function (Some continued functions, The continued fraction for \(e\), The basic asymptotic estimate).

Bibliography, Appendices A, B, C.

Reviewer’s comment. The original edition of this book was published in 1966. The conclusion of the foreword to the first edition is: “If, however, like Rip van Winkle, I should awake from slumber in twenty years, my greatest hope would be that the theory by then had acquired the broad coherence which it deserves”. Apparently, even almost 30 years have not been sufficient.

As pointed out in the foreword to this new second edition, the only change from the first is “the addition of two papers, written in collaboration with W. Adams and H. Trotter, giving computational information for the behavior of certain algebraic and classical transcendental numbers with respect to approximation by rational numbers and their continued fraction”. These papers are [W. Adams and S. Lang, Some computations in diophantine approximations, J. Reine Angew. Math. 220, 163-173 (1965; Zbl 0142.295); S. Lang and H. Trotter, J. Reine Angew. Math. 255, 112-134 (1972; Zbl 0237.10022); Addendum, ibid. 267, 219-220 (1974; Zbl 0305.10026)].

Contents: Chapter 1: General formalism (Rational continued fractions, The continued fraction of a real number, Equivalent numbers, Intermediate convergents).

Chapter II: Asymptotic approximations (Distribution of the convergents, Numbers of constant type, Asymptotic approximations, Relation with continued fractions).

Chapter III: Estimates of averaging sums (The sum of the remainders, The sum of the reciprocals, Quadratic exponential sums, Sums with more general functions).

Chapter IV: Quadratic irrationalities (Quadratic numbers of periodicity, Units and continued fractions, The basic asymptotic estimate).

Chapter V: The exponential function (Some continued functions, The continued fraction for \(e\), The basic asymptotic estimate).

Bibliography, Appendices A, B, C.

Reviewer’s comment. The original edition of this book was published in 1966. The conclusion of the foreword to the first edition is: “If, however, like Rip van Winkle, I should awake from slumber in twenty years, my greatest hope would be that the theory by then had acquired the broad coherence which it deserves”. Apparently, even almost 30 years have not been sufficient.

Reviewer: M.Waldschmidt (Paris)

##### MSC:

11Jxx | Diophantine approximation, transcendental number theory |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11J25 | Diophantine inequalities |

11J68 | Approximation to algebraic numbers |

11K60 | Diophantine approximation in probabilistic number theory |