Introduction to Diophantine Approximations

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Springer Science & Business Media, Jun 29, 1995 - Mathematics - 130 pages
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.
 

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Contents

General Formalism
1
2 The Continued Fraction of a Real Number
6
3 Equivalent Numbers
11
4Intermediate Convergents
15
Asymptotic Approximations
20
2 Numbers of Constant Type
23
3 Asymtotic Approximations
25
4 Relation with Continued Fractions
32
Quadratic Irrationalities
50
2 Units and Continued Fractions
55
3 The Basic Asymptotic Estimate
61
The Exponential Function
69
2 The Continued Fraction for e
72
3 The Basic Asymptotic Estimate
73
Bibliography
79
Some Computations in Diophantine Approximations
81

Estimates of Averaging Sums
35
2 The Sum of the Reciprocals
37
3 Quandratic Exponential Sums
41
4 Sums with More General Functions
45
Continued Fractions for Some Algebraic Numbers
93
Addendum to Continued Fractions for Some Algebraic Numbers
126
Index
129
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