Approximation by Algebraic Numbers

Author: Yann Bugeaud

Publisher: Cambridge University Press

ISBN: 9781139455671

Category: Mathematics

Page: N.A

View: 767

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Algebraic numbers can approximate and classify any real number. Here, the author gathers together results about such approximations and classifications. Written for a broad audience, the book is accessible and self-contained, with complete and detailed proofs. Starting from continued fractions and Khintchine's theorem, Bugeaud introduces a variety of techniques, ranging from explicit constructions to metric number theory, including the theory of Hausdorff dimension. So armed, the reader is led to such celebrated advanced results as the proof of Mahler's conjecture on S-numbers, the Jarnik–Besicovitch theorem, and the existence of T-numbers. Brief consideration is given both to the p-adic and the formal power series cases. Thus the book can be used for graduate courses on Diophantine approximation (some 40 exercises are supplied), or as an introduction for non-experts. Specialists will appreciate the collection of over 50 open problems and the rich and comprehensive list of more than 600 references.
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Number Theory, Analysis and Geometry

In Memory of Serge Lang

Author: Dorian Goldfeld,Jay Jorgenson,Peter Jones,Dinakar Ramakrishnan,Kenneth Ribet,John Tate

Publisher: Springer Science & Business Media

ISBN: 1461412595

Category: Mathematics

Page: 704

View: 3080

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In honor of Serge Lang’s vast contribution to mathematics, this memorial volume presents articles by prominent mathematicians. Reflecting the breadth of Lang's own interests and accomplishments, these essays span the field of Number Theory, Analysis and Geometry.
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Distribution Modulo One and Diophantine Approximation

Author: Yann Bugeaud

Publisher: Cambridge University Press

ISBN: 0521111692

Category: Mathematics

Page: 300

View: 6694

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A treatment of cutting-edge research on the distribution modulo one of sequences and related topics, much of it from the last decade. There are numerous exercises to aid student understanding of the topic, and researchers will appreciate the notes at the end of each chapter, extensive references and open problems.
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Diophantine Approximation

Festschrift for Wolfgang Schmidt

Author: Robert F. Tichy,Hans Peter Schlickewei,Klaus D. Schmidt

Publisher: Springer Science & Business Media

ISBN: 3211742808

Category: Mathematics

Page: 422

View: 2007

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This volume contains 21 research and survey papers on recent developments in the field of diophantine approximation, which are based on lectures given at a conference at the Erwin Schrödinger-Institute (Vienna, 2003). The articles are either in the spirit of more classical diophantine analysis or of a geometric or combinatorial flavor. Several articles deal with estimates for the number of solutions of diophantine equations as well as with congruences and polynomials.
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Diophantine Geometry

Proceedings

Author: Umberto Zannier

Publisher: Springer

ISBN: N.A

Category: Mathematics

Page: 390

View: 2180

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This book contains research articles on Diophantine Geometry, written by participants of a research program held at the Ennio De Giorgi Mathematical Research Center in Pisa, Italy, between April and July 2005. The authors are eminent experts in the field and present several subfields of the main topic. The volume provides a broad overview of recent research developments.
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Diophantine Approximation on Linear Algebraic Groups

Transcendence Properties of the Exponential Function in Several Variables

Author: Michel Waldschmidt

Publisher: Springer Science & Business Media

ISBN: 3662115697

Category: Mathematics

Page: 633

View: 9203

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The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. Two chapters provide complete and simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.
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Metric Diophantine Approximation on Manifolds

Author: Vasiliĭ Ivanovich Bernik,M. M. Dodson

Publisher: Cambridge University Press

ISBN: 9780521432757

Category: Mathematics

Page: 172

View: 8156

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This volume explores Diophantine approximation on smooth manifolds embedded in Euclidean space, developing a coherent body of theory comparable to that of classical Diophantine approximation. In particular, the book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. After setting out the necessary background material, the authors give a full discussion of Hausdorff dimension and its uses in Diophantine approximation. They employ a wide range of techniques from the number theory arsenal to obtain the upper and lower bounds required, highlighting the difficulty of some of the questions considered. The authors then go on to consider briefly the p-adic case, and conclude with a chapter on some applications of metric Diophantine approximation. All researchers with an interest in Diophantine approximation will want to have this book in their personal libraries.
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