Introduction to Diophantine Approximations: New Expanded Edition

Introduction to Diophantine Approximations: New Expanded Edition

by Serge Lang

Paperback(2nd ed. 1995)

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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.

Product Details

ISBN-13: 9781461287001
Publisher: Springer New York
Publication date: 08/22/2012
Edition description: 2nd ed. 1995
Pages: 130
Product dimensions: 6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

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.

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