Introduction to Diophantine Approximations - Serge Lang

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Résumé :
The aim of this book is to illustrate by significant special examples three aspects of the theory of Diophantine approximations: the formal relationships that exist between counting processes and the functions entering the theory; the determination of these functions for numbers given as classical numbers; and certain asymptotic estimates holding almost everywhere. Each chapter works out a special case of a much broader general theory, as yet unknown. Indications for this are given throughout the book, together with reference to current publications. The book may be used in a course in number theory, whose students will thus be put in contact with interesting but accessible problems on the ground floor of mathematics.

Sommaire:
I General Formalism.- ?1. Rational Continued Functions.- ?2. The Continued Fraction of a Real Number.- ?3. Equivalent Numbers.- ?4. Intermediate Convergents.- II Asymptotic Approximations.- ?1. Distribution of the Convergents.- ?2. Numbers of Constant Type.- ?3. Asymptotic Approximations.- ?4. Relation with Continued Fractions.- III Estimates of Averaging Sums.- ?1. The Sum of the Remainders.- ?2. The Sum of the Reciprocals.- ?3. Quadratic Exponential Sums.- ?4. Sums with More General Functions.- IV Quadratic Irrationalities.- ?1. Quadratic Numbers and Periodicity.- ?2. Units and Continued Fractions.- ?3. The Basic Asymptotic Estimate.- V The Exponential Function.- ?1. Some Continued Functions.- ?2. The Continued Fraction for e.- ?3. The Basic Asymptotic Estimate.- Appendix A Some Computations in Diophantine Approximations.- Appendix B Continued Fractions for Some Algebraic Numbers.- Appendix C Addendum to Continued Fractions for Some Algebraic Numbers.