Introduction to Diophantine Approximations New Expanded Edition [Hardcover]

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.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.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 Irrationalitil¢