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Normal Numbers with Respect to the Cantor Series Expansion

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Normal Numbers with Respect to the Cantor Series Expansion NORMAL NUMBERS WITH RESPECT TO THE CANTOR SERIES EXPANSION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Bill Mance, M.S. Graduate Program in Mathematics The Ohio State University 2010 Dissertation Committee: Dr. Vitaly Bergelson, Advisor Dr. Gerald Edgar Dr. Alexander Leibman c Copyright by Bill Mance 2010 ABSTRACT 1 A sequence Q = fqngn=1 is called a basic sequence if each qn is an integer greater than or equal to 2. A basic sequence is infinite in limit if qn ! 1. The Q-Cantor series expansion, first studied by G. Cantor, is a generalization of the b-ary expansion P1 En where every real number in [0; 1) is expressed in the form with En 2 n=1 q1q2:::qn [0; qn − 1] \ Z and En 6= qn − 1 infinitely often. A real number x is normal in base b if every block of digits of length k occurs with frequency b−k in its b-ary expansion; n 1 equivalently, the sequence fb xgn=0 is uniformly distributed mod 1. The notion of normality is extended to the Q-Cantor series expansion. We primar- ily consider three distinct notions of normality that are equivalent in the case of the b-ary expansion: Q-normality, Q-ratio normality, and Q-distribution normality. All Q-normal numbers are Q-ratio normal, but there is no inclusion between Q-normal numbers and Q-distribution normal numbers. Thus, the fundamental equivalence between notions of normality that holds for the b-ary expansion will no longer hold for the Q-Cantor series expansion, depending on the basic sequence Q. We prove theorems that may be used to construct Q-normal and Q-distribution normal numbers for a restricted class of basic sequences Q. Using these theorems, we construct a number that is simultaneously Q-normal and Q-distribution normal for a certain Q. We also use the same theorems to provide an example of a basic ii sequence Q and a number that is Q-normal, yet fails to be Q-distribution normal in a particularly strong manner. Many constructions of numbers that are Q-distribution normal, yet not Q-ratio normal are also provided. In [24], P. Laffer asked for a construction of a Q-distribution normal number given an arbitrary Q. We provide a partial answer by constructing an uncountable family 1 of Q-distribution normal numbers, provided that Q = fqngn=1 satisfies the condition that it is infinite in limit. This set of Q-distribution normal numbers that we construct has the additional property that it is perfect and nowhere dense. Additionally, none of these numbers will be Q-ratio normal. Also studied are questions of typicality for different notions of normality. We show that under certain conditions on the basic sequence Q, almost every real number is Q-normal. If Q is infinite in limit, then the set of Q-ratio normal numbers will be dense in [0; 1), but may or may not have full measure. Almost every real number will be Q-distribution normal no matter our choice of Q. The set of Q-ratio normal and the set of Q-distribution normal numbers are small in the topological sense; they are both sets of the first category. We also study topological properties of other sets relating to digits of the Q-Cantor series expansion. We define potentially stronger notions of normality: strong Q-normality, strong Q- ratio normality, and strong Q-distribution normality that are equivalent to normality in the case of the b-ary expansion. We show that the set of strongly Q-distribution normal numbers always has full measure, but the set of strongly Q-normal numbers will only under certain conditions. We study winning sets, in the sense of Schmidt games and show that the set of non-strongly Q-ratio normal numbers and the set iii of non-strongly Q-distribution normal numbers are 1=2-winning sets and thus have full Hausdorff dimension. We also examine the property of being a winning set as it applies to other sets associated with the Q-Cantor series expansion. A number normal in base b is never rational. We study how well this notion transfers to the Q-Cantor series expansion. In particular, it will remain consistent for Q-distribution normal numbers, but fail in unusual ways for other notions of normality. iv Dedicated to Abigail. v ACKNOWLEDGMENTS First and foremost, I wish to thank Vitaly Bergelson, my advisor. No one else has had a greater mathematical influence over me. I am indebted to him for guiding me during my time at The Ohio State University and pointing me towards my current path of research. No one can be expected to give me as much time and energy as he has and for this I am truly grateful. Second, I would like to thank everyone at the Indus Center for Academic Excel- lence, especially Raghunath Khetan, for helping me develop my problem solving skills early in my mathematical career. Third, I wish to thank Gerald Edgar and Alexander Leibman for their time and effort participating in my thesis committee. I wish to thank Mary Leary of Bishop Foley Catholic High School for allowing and encouraging me to explore mathematics on my own when I was in high school. I would also like to thank Ruth Favro of Lawrence Technological University for providing guidance in my pre-college years. I would like to thank Christian Altomare for the many valuable discussions. Ad- ditionally, I would like to thank Jim Tseng whose talk \Nondense orbits, symbolic dynamics, and games", given in the Ergodic Theory and Probability Seminar at The Ohio State University in October 2008, led me to invistigate Schmidt's game and how vi it pertains to sets of non-normal numbers. This ultimately formed the basis for one of the chapters of this dissertation. I wish to thank Cindy Bernlohr for help and guidance as a teacher during my time at Ohio State. I would also like to thank everyone else at Carnegie Mellon University and The Ohio State University who contributed to my understanding of mathematics or helped me out in any way. I wish to thank Doug Heath for invaluable guidance as a powerlifter and for letting me train in his basement. I never would have gotten as far as I have without his influence. Last but not least, I thank my parents, Andrew and Susan Mance. vii VITA April 6, 1981 . Born - Royal Oak, MI 1999-2003 . Undergraduate, Carnegie Mellon University 2003 . B.S. in Mathematics, Carnegie Mellon University 2003-Present . Graduate Teaching Associate, The Ohio State University 2004 . M.S. in Mathematics, The Ohio State University PUBLICATIONS Construction of normal numbers with respect to the Q-Cantor series expansion for certain Q, arXiv:0911.1485v1 (preprint). Cantor series constructions contrasting two notions of normality (with C. Altomare), Monatsh. Math. (to appear). viii FIELDS OF STUDY Major Field: Mathematics Specialization: Normal Numbers and Uniform Distribution of Sequences ix TABLE OF CONTENTS Abstract . ii Dedication . v Acknowledgments . vi Vita . viii CHAPTER PAGE 1 Introduction . 1 1.1 E. Borel and Early Developments in the Theory of Normal Numbers . 1 1.2 Uniformly Distributed Sequences . 5 1.2.1 Connection to Normality . 6 1.2.2 Classical Results . 7 1.2.3 Discrepancy . 9 1.2.4 Examples of Uniformly Distributed Sequences . 10 1.3 Ergodic Theory and Common Series Expansions . 12 1.3.1 Definitions and Basic Examples . 12 1.3.2 A Look at the b-ary Expansion Through Ergodic Theory . 14 1.3.3 The Continued Fraction Expansion . 15 1.3.4 The L¨urothSeries Expansion . 19 1.3.5 The Generalized L¨urothSeries Expansion . 21 1.3.6 β-expansions . 23 1.4 f-expansions . 25 1.5 Other Common Expansions . 29 1.5.1 The Engel Series Expansion . 29 1.5.2 The Sylvester Series Expansion . 31 1.5.3 The Cantor Product . 33 1.6 Fundamental Properties of Normal Numbers . 34 x 2 Normality With Respect to the Cantor Series Expansion . 36 2.1 The Cantor Series Expansion . 36 2.2 Basic Definitions Relating to Normality . 39 2.3 Q-Normal Numbers . 43 2.4 Q-Distribution Normal Numbers . 44 2.5 Q-Ratio Normal Numbers . 51 2.6 Basic Examples . 54 2.6.1 Q-Distribution Normality Without Simple Q-Normality . 54 2.6.2 A Simply Q-Normal Number for a 1-Convergent Q . 55 2.6.3 Example of a Number that is Simply Q-Normal and Q-Distribution Normal for a Non-Trivial Basic Sequence Q . 56 2.6.4 A Simply Q-Ratio Normal Number that is not Simply Q-Normal or Q-Distribution Normal . 61 3 General Construction Theorems for Q-Normal and Q-Distribution Nor- mal Numbers for Certain Non-Trivial Q . 63 3.1 Basic Definitions and Conventions . 63 3.2 Modular Friendly Families and Construction of Q-Distribution Normal Numbers . 66 3.2.1 MFFs . 66 3.2.2 Discrepancy and V -Nice Sequences . 68 3.2.3 Main Theorem . 78 3.3 Block Friendly Families and Construction of Q-Normal Numbers . 80 3.3.1 BFFs . 80 3.3.2 Technical Lemmas . 81 3.3.3 Main Theorem . 98 4 Construction of a Number that is Q-Normal and Q-Distribution Normal for a Certain Non-Trivial Q . 100 5 Construction of a Number that is Q-Normal and not Q-Distribution Normal . 107 6 Construction of Q-Distribution Normal Numbers for Arbitrary Q that are Infinite in Limit .
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