A structure theorem on doubling measures with different bases

TL;DR

This paper establishes a structure theorem for unions of n-adic doubling measures using advanced techniques, extending Wu's 1998 results and also providing a classification related to normal numbers.

Contribution

It introduces a novel structure theorem for infinite unions of n-adic doubling measures and extends prior work by Wu, also linking to normal number classification.

Findings

Proved a structure theorem for unions of n-adic doubling measures.

Extended Wu's 1998 results on doubling measures.

Provided a classification result related to normal numbers.

Abstract

In this paper, we prove a structure theorem for the infinite union of $n$-adic doubling measures via techniques which involve far numbers. Our approach extends the results of Wu in 1998, and as a by product, we also prove a classification result related to normal numbers.