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 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.
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.
Inhaltsverzeichnis
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.
Klappentext
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.
