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AuthorLang, Serge. author
TitleIntroduction to Diophantine Approximations [electronic resource] : New Expanded Edition / by Serge Lang
ImprintNew York, NY : Springer New York, 1995
Connect tohttp://dx.doi.org/10.1007/978-1-4612-4220-8
Descript X, 130 p. online resource

SUMMARY

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


CONTENT

I General Formalism -- ยง1. Rational Continued Functions -- ยง2. The Continued Fraction of a Real Number -- ยง3. Equivalent Numbers -- ยง4. Intermediate Convergents -- II Asymptotic Approximations -- ยง1. Distribution of the Convergents -- ยง2. Numbers of Constant Type -- ยง3. Asymptotic Approximations -- ยง4. Relation with Continued Fractions -- III Estimates of Averaging Sums -- ยง1. The Sum of the Remainders -- ยง2. The Sum of the Reciprocals -- ยง3. Quadratic Exponential Sums -- ยง4. Sums with More General Functions -- IV Quadratic Irrationalities -- ยง1. Quadratic Numbers and Periodicity -- ยง2. Units and Continued Fractions -- ยง3. The Basic Asymptotic Estimate -- V The Exponential Function -- ยง1. Some Continued Functions -- ยง2. The Continued Fraction for e -- ยง3. 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


Mathematics Number theory Mathematics Number Theory



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