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Normal Numbers with Respect to the Cantor Series Expansion
Mance, Bill

2010, Doctor of Philosophy, Ohio State University, Mathematics.

A sequence $Q={q_n}_{n=1}^{infty}$ 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 $q_n → infty$. The Q-Cantor series expansion, first studied by G. Cantor, is a generalization of the b-ary expansion where every real number in $[0,1)$ is expressed in the form sum_{n=1}^{infty} frac {E_n} {q_1 q_2 ldots q_n} with $E_n in [0,q_n-1] cap mathbb{Z}$ and $E_n
eq q_n-1$ 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; equivalently, the sequence {b^n x}_{n=0}^{infty} is uniformly distributed mod 1.

The notion of normality is extended to the Q-Cantor series expansion. We primarily 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 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 cite{Laffer}, 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 of Q-distribution normal numbers, provided that $Q={q_n}_{n=1}^{infty}$ 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 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.

Vitaly Bergelson, PhD (Advisor)
Gerald Edgar, PhD (Committee Member)
Alexander Leibman, PhD (Committee Member)
Luis Rademacher, PhD (Committee Member)
290 p.

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Mance, B. (2010). Normal Numbers with Respect to the Cantor Series Expansion. (Electronic Thesis or Dissertation). Retrieved from

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Mance, Bill. "Normal Numbers with Respect to the Cantor Series Expansion." Electronic Thesis or Dissertation. Ohio State University, 2010. OhioLINK Electronic Theses and Dissertations Center. 26 Sep 2017.

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Mance, Bill "Normal Numbers with Respect to the Cantor Series Expansion." Electronic Thesis or Dissertation. Ohio State University, 2010.


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