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.