Descriptive Complexity in Cantor Series

Abstract

A Cantor series expansion for a real number x with respect to a basic sequence , where , is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a -complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is -complete, and if Q is -divergent then the sets of normal and ratio normal numbers are -complete. We further show that all five non-trivial differences of these sets are -complete if and Q is -divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible.