Normality of different orders for Cantor series expansions

Abstract

Let have the property that for each the set has asymptotic density 0. We prove that there exists a basic sequence Q where the set of numbers Q-normal of all orders in S but not Q-normal of all orders not in S has full Hausdorff dimension. If the function is computable, then there exist computable examples. For example, there exists a computable basic sequence Q where the set of numbers normal of all even orders and not normal of all odd orders has full Hausdorff dimension. This is in strong constrast to the b-ary expansions where any real number that is normal of order k must also be normal of all orders between 1 and .Additionally, all numbers we construct satisfy the unusual condition that block frequencies sampled along non-trivial arithmetic progressions do not converge to the expected value. This is also in strong contrast to the case of the b-ary expansions, but more similar to the case of the continued fraction expansion. As a corollary, the set of Q-normal numbers that are not normal when sampled along any non-trivial arithmetic progression has full Hausdorff dimension.