As a special case of a theorem of Pollington, the set of numbers that are normal in base $2$, but not normal in base $3$ is known to have full Hausdorff dimension. An interpretation of results such as this is that in some way the notions of normality in base $2$ and normality in base $3$ are "independent".

The purpose of this talk is to introduce notions from descriptive set theory that may better capture this idea of independence: that of $D_2(\Sigma^0_\alpha)$- and $D_2(\Pi^0_\alpha)$-completeness. For example, the previously mentioned set of numbers normal in base $2$, but not normal in base $3$ is known to be $D_2(\Pi^0_\alpha)$-complete by a recent result of Jackson, M., and Vandehey. Difference sets that satisfy these properties will also have additional interesting properties that will be discussed.