Diophantine Approximation
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B. Eckmann, ZUrich
785
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Wolfgang M. Schmidt
Diophantine Approximation
Springer
Author Wolfgang M. Schmidt Department of Mathematics University of Colorado Boulder, CO 80309, USA
1st edition 1980 2nd printing 1996 (with minor corrections)
Mathematics Subject Classification (1980): 10B16, 10E05, 10E15, 10E40, 10F05, 10FI0, 10F20, 10F30, 10K15
Library of Congress Cataloging in Publication Data. Schmidt, Wolfgang M., Diophantine approximation. (Lecture notes in mathematics; 785) Bibliography: p. Includes index. 1. Algebraic number theory. 2.Approximation, Diophantine. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 785. QA3.L28 no. 785 [QA247] 510s [512' .74] 80-11695 ISBN 0-387-09762-7
ISBN 3-540-09762-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, were copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© Springer-Verlag Berlin Heidelberg 1980 Printed in Germany SPIN: 10530138
46/2142-543210 - Printed on acid-free paper
Preface In spring 1970 I gave a course in Diophantine Approximation at the University of Colorado, which culminated in simultaneous approximation to algebraic numbers. soon gone.
A limited supply of mimeographed Lecture Notes was
The completion of these new Notes was greatly delayed by my
decision to add further material. The present chapter on sinultaneous approximations to algebraic numbers is much more general than the one in the original Notes.
This
generality is necessary to supply a basis for the subsequent chapter on norm form equations. algebraic numbers.
There is a new last chapter on approximation by I wish to thank all those, in particular Professor
C.L. Siegel, who have pointed out a number of mistakes in the original Notes.
I hope that not too many new mistakes have crept into these new
Notes. The present Notes contain only a small part of the theory of Diophantine Approximation.. algebraic numbers.
The main emphasis is on approximation to
But even here not everything is included.
I follow
the approach which was initiated by Thue in 1908, and further developed by Siegel and by Roth, but I do not include the effective results due to Baker.
Not included is approximation
in
p - adic fields, for which
see e.g. Schlickewei [1976, 1977], or approximation fields, for which
s~e
in power series
e.g., Osgood [1977] and Ratliff [19781.
Totally
missing are Pisot-Vijayaraghavan Numbers, inhomogeneous approximation and uniform distribution.
For these see e.g. Cassels [1957] and Kuipers
and Niederreiter [1974].
Also excluded a
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