We will discuss extending the concept of normality to the $Q$-Cantor series expansions by defining two notions that are equivalent for $b$-ary expansions: $Q$-normality and $Q$-distribution normality. Almost every real number is $Q$-distribution normal and the sets of non-$Q$-normal and non-$Q$-distribution normal numbers are residual sets with full Hausdorff dimension. Furthermore, these sets are even $1/2$-winning sets. Surprisingly, $Q$-normality and $Q$-distribution normality are no longer equivalent. We will provide recent constructions that demonstrate this. In fact, there is reason to suspect that the set of numbers that is $Q$-distribution normal but not $Q$-normal has full Hausdorff dimension. Other open questions will also be presented.

This talk is the first part in a series of presentations that will be continued in the RTG Logic and Dynamics Seminar.