Metric Diophantine Approximation on Manifolds

by V. I. Bernik, M. M. Dodson

Overview

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.

Product Details

ISBN-13:	9780521432757
Publisher:	Cambridge University Press
Publication date:	10/14/1999
Series:	Cambridge Tracts in Mathematics , #137
Pages:	186
Product dimensions:	5.98(w) x 9.02(h) x 0.55(d)

Table of Contents

1. Diophantine approximation; 2. Khintchine-type manifolds; 3. Hausdorff measure and dimension; 4. Upper bounds; 5. Lower bounds for Hausdorff dimension; 6. p-adic Diophantine approximation; 7. Applications.

Editorial Reviews

"The book represents an undispensable tool for those who work in Diophantine Approximation." Monatshefte fur Mathematik

From the Publisher

In order to develop a coherent body of theory along the lines of the existing one for the classical theory, for manifolds that are Euclidean space, Bernik (Byelorussian Academy of Sciences, Minsk) and Dodson (U. of York, England) explore metric Diophantine approximations on smooth manifolds embedded in Euclidian space. They warn that the functional dependence of the coordinates presents serious technical difficulties, but promise a surprising degree of interplay between the very different areas of number theory, differential geometry, and measure theory. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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