A structure theorem on doubling measures with different bases: a number theoretic approach

In seemingly unrelated statements, yet interconnected proofs, we provide two distinct structural classifications related to normal numbers and $n$-adic doubling measures. The motivation for this is originally due to refined questions about unions of certain doubling measures important in analysis; in particular we are able to classify $n$-adic doubling measures in a way that extends results of Wu from a single measure to an infinite union. However, the proof strategy that we employ is completely different from Wu's and heavily utilizes many number theoretic elements, which leads to the second, unexpected, classification related to normal numbers.