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Uniform Distribution of Sequences Alt 1-4 'A I #so 4, I s4.1 L" - _.u or- ''Ifi r '. ,- I 'may,, i ! :' m UNIFORM DISTRIHUTION OF SEQUENCES L. KUIPERS Southern Illinois University H. NIEDERREITER Southern Illinois University A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS New York London Sydney Toronto Copyright © 1974, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language with- out the written permission of the publisher. Library of Congress Cataloging in Publication Data: Kuipers, Lauwerens. Uniform distribution of sequences. (Pure and applied mathematics) "A Wiley-Interscience publication." Bibliography: 1. Sequences (Mathematics) 1. Niederreiter, H., joint author. II. Title. QA292.K84 515'.242 73-20497 ISBN 0-471-51045-9 Printed in the United States of America 10-987654321 To Francina and Gerlinde REFACCE The theory of uniform distribution modulo one (Gleichverteilung modulo Eins, equirepartition modulo un) is concerned, at least in its classical setting, with the distribution of fractional parts of real numbers in the unit interval (0, 1). The development of this theory started with Hermann Weyl's celebrated paper of 1916 titled: "Uber die Gleichverteilung von Zahlen mod. Eins." Weyl's work was primarily intended as a refinement of Kronecker's approxi- mation theorem, and, therefore, in its initial stage, the theory was deeply rooted in diophantine approximations. During the last decades the theory has unfolded beyond that framework. Today, the subject presents itself as a meeting ground for topics as diverse as number theory, probability theory, functional analysis, topological algebra, and so on. However, it must be said that the germs of various later developments can be found in Weyl's 1916 paper. This book attempts to summarize the results of these investigations from the beginning to the present, with emphasis on the work done during the last 20 years. Because the literature on the subject is vast, it was inevitable that now and then choices had to be made and selection criteria had to be applied that reflect the personal taste of the authors. As a rule, we have endeavored to produce a comprehensive coverage of the methods used in the theory of uniform distribution. In some instances, we did not present a result in its most general version, but rather tried to describe the underlying principles and ideas, which might otherwise be shrouded in technicalities. The title we have chosen indicates that we have resolved the dilemma of "asymptotic distribution" versus "uniform distribution" in favor of the latter, since it is this aspect to which most of our exposition is devoted. We believe that this book should prove a useful introduction to the subject for students in number theory and analysis and a reference source for researchers in the field. An important role in our presentation is played by the notes at the end of each section. These not only contain the pertinent bibliographical refer- ences, but also provide the reader with a brief survey of additional results relating to the material of that section. The exercises we include range from simple applications of theorems to proofs of propositions more general vii viii PREFACE than, or shedding further light on, results in the text. The reader is encouraged to try his hand at many of these problems in order to increase his under- standing of the theory. The following is a brief outline of the contents of the book. The first chapter deals with the classical part of the theory. It assumes that the reader has a good background in real analysis. Important properties of uniformly distributed sequences of real numbers are developed in the early sections and specific examples of uniformly distributed sequences are described through- out. A central position in the theory is taken by the so-called Weyl criterion. As a means for investigating sequences with respect to uniform distribution, it caused in the early years of the development of the theory of uniform distribution a strong interest in exponential sums. One of the equivalents of the definition of a uniformly distributed sequence modulo 1 of real numbers is the functional definition. This form of the notion of uniform distribution reveals the measure-theoretic and topological character of the definition. To avoid repetition, we have, in the first chapter, concentrated on techniques and results that depend on the special structure of the real number system. Results that have analogues in a general setting (such as van der Corput's difference theorem and various metric theorems) will often be found in stronger versions in the more abstract Chapters 3 and 4. Some extensions of the theory are already touched on in Chapter 1, such as the multidimensional case in Section 6 and distribution functions and asymptotic distribution with respect to summation methods in Section 7. A study of normal numbers and their relation to uniform distribution is carried out in Section 8. Uniform distribution modulo 1 of measurable functions appears in Section 9. In Chapter 2, the results of the preceding chapter are studied and comple- mented from a quantitative point of view. Here a modest background in number theory would be helpful. Many of the results presented here are current. The important results of K. F. Roth and W. M. Schmidt on ir- regularities of distribution, as well as the celebrated inequalities of Erdos- Turan and LeVeque, are proved in Section 2. The next section is concerned mainly with the sequences (n6), n = 1, 2...,with 0 irrational. This leads one back to the number-theoretic origins of the theory. Number-theoretic integration methods which derive from the theory of uniform distribution are treated in Section 5. This part of the book should be of interest to numerical analysts. In Chapters 3 and 4, we develop in greater detail the theory of uniform distribution in compact Hausdorff spaces and in topological groups, re- spectively. A background in topology and measure theory is required. Furthermore, for Chapter 4 a knowledge of topological groups is desirable, but all the required results from structure theory and duality theory are stated. Many interesting relations to probability theory, ergodic theory, PREFACE ix summability theory, and topological algebra emerge in these portions of the book. In Section 2 of Chapter 4 a detailed treatment of the method of correlation functions is given. Section 4 of Chapter 4 contains an in-depth discussion of the theory of monothetic groups. The most recent branch of the theory is based on the notion of a uniformly distributed sequence of rational integers. It is not surprising that, sub- sequently, attention has been paid to the distribution of sequences of a more general type of integers, such as the p-adic ones, and of sequences of elements of the ring of polynomials over a finite field. These developments are described in Chapter 5. The reader should be familiar with the rudiments of p-adic number theory and the theory of finite fields. Because of space limitations a number of topics could not be covered as fully as we wished or had to be omitted completely in our presentation. Topics in the latter category include metric quantitative results, relations between uniform distribution and harmonic analysis, and the theory of weak con- vergence of probability measures. However, the notes in the appropriate sections contain a survey of the literature on these aspects. A graduate course in uniform distribution emphasizing the number- theoretic connections could be based on Chapters 1, 2, and 5 of the book and may be complemented by selections from the remaining chapters. The prerequisites for each chapter have been mentioned above. It is with great pleasure and gratitude that we acknowledge helpful con- versations and/or correspondence with I. D. Berg, D. L. Carlson, J. Cigler, S. Haber, J. H. Halton, W. J. LeVeque, H. G. Meijer, L. A. Rubel, W. M. Schmidt, R. Tijdeman, and S. K. Zaremba. Special thanks are due Professor Walter Philipp, who read drafts of the manuscript and made a number of valuable suggestions. The second author would like to express his in- debtedness to Professor Edmund Hlawka, who guided his first steps into the theory of uniform distribution and whose supreme mastery of the subject was a constant source of enlightenment for his students. He also wants to thank the University of Illinois for its hospitality during a crucial period of the writing. The enormous task of converting our drafts into a typewritten manuscript was mastered by Miss Pat Coombs, whose skill and unfailing patience we learned to admire. We extend our sincere gratitude to the staff of Wiley- Interscience for its care and professionalism in the production of the book. October 1973 L. KUIPERS H. NIEDERREITER CONTENTS Chapter 1Uniform Distribution Mod 1 1 1.Definition 1 Uniform distribution modulo 1 1, Uniform distribution modulo a subdivision 4, Notes 5, Exercises 6. 2.The Weyl criterion7 Thecriterion7,Applicationsto special sequences 8, Applications to power series 10, Fejer's theorem 13, An estimate for exponential sums 15,Uniform distribution of double sequences 18, Notes 21, Exer- cises 23. 3.Difference theorems25 Van der Corput's difference theorem 25, Other differ- ence theorems 28, Notes 30, Exercises 30. 4.Metric theorems32 Some basic results32,Koksma's general metric theorem 34, Trigonometric sequences 36, Notes 39, Exercises 40. 5.Well-distributed sequences mod 140 Definition and Weyl criteria 40, Admissible sequences 42, Metric theorems 44, Notes 46, Exercises 46. 6.The multidimensional case47 Definitionand basicresults47,Applications 50, Notes 51, Exercises 51. 7.Distribution functions53 Various types of distribution functions 53, Criteria 54, Miscellaneous results 55, An elementary method 57, xi xii CONTENTS Metric theorems 59, Summation methods 60, Notes 66, Exercises 67.
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