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Ergodic Theory of Numbers AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 29 ERGODIC THEORY OF NUMBERS KARMA DAJANI COR KRAAIKAMP Ergodic Theory of Numbers © 2002 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2002101376 Print ISBN 978-0-88385-034-3 eISBN 978-1-61444-027-7 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 3 2 1 10.1090/car/029 The Carus Mathematical Monographs Number Twenty-Nine Ergodic Theory of Numbers Karma Dajani University of Utrecht Cor Kraaikamp Delft University of Technology Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications Gerald L. Alexanderson, Chair Carus Mathematical Monographs Editorial Board Kenneth A. Ross, Editor Joseph Auslander Harold P. Boas Robert E. Greene Dennis Hejhal Roger Horn Jeffrey Lagarias Barbara Osofsky Clarence Eugene Wayne David Wright The following Monographs have been published: 1. Calculus of Variations, by G. A. Bliss (out of print) 2. Analytic Functions of a Complex Variable, by D. R. Curtiss (out of print) 3. Mathematical Statistics, by H. L. Rietz (out of print) 4. Projective Geometry, by J. W. Young (out of print) 5. A History of Mathematics in America before 1900, by D. E. Smith and Jekuthiel Ginsburg (out of print) 6. Fourier Series and Orthogonal Polynomials, by Dunham Jackson (out of print) 7. Vectors and Matrices, by C. C. MacDuffee (out of print) 8. Rings and Ideals, by N. H. McCoy (out of print) 9. The Theory of Algebraic Numbers, second edition, by Harry Pollard and Harold G. Diamond 10. The Arithmetic Theory of Quadratic Forms, by B. W. Jones (out of print) 11. Irrational Numbers, by Ivan Niven 12. Statistical Independence in Probability, Analysis and Number Theory, by Mark Kac 13. A Primer of Real Functions, third edition, by Ralph P. Boas, Jr. 14. Combinatorial Mathematics, by Herbert J. Ryser 15. Noncommutative Rings, by I. N. Herstein (out of print) 16. Dedekind Sums, by Hans Rademacher and Emil Grosswald 17. The Schwarz Function and Its Applications, by Philip J. Davis 18. Celestial Mechanics, by Harry Pollard 19. Field Theory and Its Classical Problems, by Charles Robert Hadlock 20. The Generalized Riemann Integral, by Robert M. McLeod 21. From Error-Correcting Codes through Sphere Packings to Simple Groups, by Thomas M. Thompson 22. Random Walks and Electric Networks, by Peter G. Doyle and J. Laurie Snell 23. Complex Analysis: The Geometric Viewpoint, by Steven G. Krantz 24. Knot Theory, by Charles Livingston 25. Algebra and Tiling: Homomorphisms in the Service of Geometry, by Sherman Stein and Sandor´ Szabo´ 26. The Sensual (Quadratic) Form, by John H. Conway assisted by Francis Y. C. Fung 27. A Panorama of Harmonic Analysis, by Steven G. Krantz 28. Inequalities from Complex Analysis, by John P. D’Angelo 29. Ergodic Theory of Numbers, by Karma Dajani and Cor Kraaikamp MAA Service Center P. O. Box 91112 Washington, DC 20090-1112 800-331-1MAA FAX: 301-206-9789 To Rafael Contents 1 Introduction 1 1.1 Decimal expansions of rational numbers . 2 1.2 Another look at the decimal expansion . 5 1.3 Continued fractions . .............. 20 1.4 For further reading .................................. 31 2 Variations on a theme (Other expansions) 33 2.1 n-ary expansions . .................... 33 2.2 Lurothseries........................................¨ 36 2.3 Generalized Lurothseries............................¨ 41 2.4 β-expansions . ...................... 51 3 Ergodicity 57 3.1 The Ergodic Theorem . ............................ 57 3.2 Examplesofnormalnumbers......................... 71 3.3 β-transformations . .................. 73 3.4 Ergodic properties of the β-expansion................. 76 3.5 Ergodic properties of continued fractions .............. 80 4 Systems obtained from other systems 89 4.1 GLS-expansion and β-expansion: A first glimpse at their connection . ............. 89 4.2 Induced and integral transformations . ................. 92 vii viii Contents 4.3 Natural extensions. ........... 98 4.4 Natural extension of the GLS transformation . .......101 4.5 Natural extension of the β-transformation . ..... 104 4.6 For further reading . ............................112 5 Diophantine approximation and continued fractions 115 5.1 Introduction . ................115 5.2 The natural extension of the regular continued fraction . 125 5.3 Approximation coefficients revisited . 126 5.4 Other continued fractions ............................134 5.5 A skew product related to continued fractions . 151 5.6 For further reading . ............................154 6 Entropy 155 6.1 Introduction . ................155 6.2 The Shannon-McMillan-Breiman Theorem and someconsequences..................................169 6.3 Saleski’sTheorem...................................175 6.4 For further reading . ............................177 Bibliography 179 Index 187 Preface In this book we will look at the interaction between two fields of math- ematics: number theory and ergodic theory (as part of dynamical sys- tems). The subject under study is thus part of what is known in France as Theorie´ Ergodique des Nombres, and consists of a family of series expansions of numbers in the unit interval [0, 1] with their ‘metrical properties.’ So the questions we want to study are number theoretical in nature, and the answers will be obtained with the help of ergodic theory. That is, we will view these expansions as iterations of an ap- propriate measure-preserving transformation on [0,1], which will then be shown to be ergodic. The number-theoretical questions will be refor- mulated in the language of ergodic theory. What it means to be ergodic, or—in general—what the basic ideas behind ergodic theory entail, will be explained along the way. This book grew out of a course given in 1996 at George Wash- ington University, Washington, DC, during the Summer Program for Women in Mathematics, sponsored by NSA. Our aim was not to write yet another book on ergodic theory (there are already several outstand- ing books, most of them mentioned in these pages), but to introduce first-year graduate students to a dynamical way of thinking. Conse- quently, many classical concepts from ergodic theory are either briefly mentioned, or even left out. In this book we focus our attention on easy concepts like ergodicity and the ergodic theorem, and then apply these ix x Preface concepts to familiar expansions to obtain old and new results in an ele- gant and straightforward manner. Clearly this means that a number of concepts from probability and measure theory will be used. In our set-up we first introduce these, in— we hope and think—an informal and gentle way. We thank the directors Murli M. Gupta, Robbie Robinson and Dan Ullman of the Summer Program for Women in Mathematics, and the participants Alissa Andreichuk, Christine Collier, Amy Cottrell, Lisa Darlington, Christy Dorman, Julie Frohlich, Molly Kovaka, Renee Yong, Ran Liu, Susan Matthews, Gail Persons, Jakayla Robbins, Beth Samuels, Elizabeth Trageser, Sharon Tyree, Meta Voelker and Joyce Williams of this summer program, who were faced with a preliminary version of this book. Their remarks, comments, improvements and en- thusiasm helped us tremendously to improve the original notes. We also thank Ken Ross and Harold Boas of the MAA, whose con- structive criticism, sharp observations and patience changed the origi- nal manuscript into a readable text. Karma Dajani Utrecht, The Netherlands Cor Kraaikamp Delft, The Netherlands Bibliography [Ada79] William W. Adams, On a relationship between the convergents of the nearest integer and regular continued fractions,Math.Comp.33 (1979), no. 148, 1321–1331. [AKS81] Roy Adler, Michael Keane, and Meir Smorodinsky, A construction of a normal number for the continued fraction transformation,J.Number Theory 13 (1981), no. 1, 95–105. [Apo90] Tom M. Apostol, Modular functions and Dirichlet series in number theory, second ed., Springer-Verlag, New York, 1990. [AD79] J. Auslander and Y.N. Dowker, On disjointness of dynamical systems, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 3, 477–491. [Bab78] K.I. Babenko, A problem of Gauss, Dokl. Akad. Nauk SSSR 238 (1978), no. 5, 1021–1024. [BM66] F. Bagemihl and J. R. McLaughlin, Generalization of some classical theorems concerning triples of consecutive convergents to simple contin- ued fractions, J. Reine Angew. Math. 221 (1966), 146–149. [BJ94] Dominique Barbolosi and Hendrik Jager, On a theorem of Legendre in the theory of continued fractions,J.Theor.´ Nombres Bordeaux 6 (1994), no. 1, 81–94. [BBDK96] Jose Barrionuevo, Robert M. Burton, Karma Dajani, and Cor Kraaikamp, Ergodic properties of generalized Luroth¨ series, Acta Arith. 74 (1996), no. 4, 311–327. [Ber77] Anne Bertrand, Developpements´ en base de Pisot et repartition´ mod- ulo 1, C. R. Acad. Sci. Paris Ser.´ A-B 285 (1977), no. 6, A419–A421. [BM96] Anne Bertrand-Mathis, Nombres normaux,J.Theor.´ Nombres Bor- deaux 8 (1996), no. 2, 397–412. 179 180 Bibliography [Bil65] Patrick Billingsley, Ergodic theory and information, John Wiley & Sons Inc., New York, 1965. [Bil95] Patrick Billingsley, Probability and measure, third ed., John Wiley & Sons Inc., New York, 1995. [Bla89] F. Blanchard, β-expansions and symbolic dynamics, Theoret. Comput. Sci. 65 (1989), no. 2, 131–141. [BJW83] W. Bosma, H. Jager, and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 3, 281–299. [BK90] Wieb Bosma and Cor Kraaikamp, Metrical theory for optimal contin- ued fractions, J. Number Theory 34 (1990), no. 3, 251–270. [BK91] Wieb Bosma and Cor Kraaikamp, Optimal approximation by contin- ued fractions, J. Austral. Math. Soc. Ser. A 50 (1991), no. 3, 481–504. [Boy96] David W. Boyd, On the beta expansion for Salem numbers of degree 6, Math. Comp. 65 (1996), no. 214, 861–875, S29–S31. [Bra51] Alfred Brauer, On algebraic equations with all but one root in the interior of the unit circle, Math. Nachr. 4 (1951), 250–257. [Bre89] David M. Bressoud, Factorization and primality testing, Springer- Verlag, New York, 1989.
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