Super-Dense Sets and Their Role in the Theory of Normal Numbers

Abstract

We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous function $\varphi\colon I\to\mathbb{R}$, we have $\varphi(I\cap A)\cap A\neq\emptyset$. We first establish basic properties of super-dense sets. Our main topological result characterizes them within the framework of Baire category: a set with the Baire property is super-dense if and only if it is co-meager. We then investigate the implications for the theory of normal numbers. We prove that the set of non-normal numbers is super-dense, whereas the set of normal numbers is not. Consequently, no nowhere constant continuous function can map all non-normal numbers to normal numbers. Conversely, we explicitly construct a computable nowhere constant continuous function that maps all normal numbers to non-normal numbers. Finally, we provide a constructive algorithm that, given any countable family of nowhere constant continuous functions, produces a real number $x$ such that $x$ and all its images under these functions are non-normal. As a corollary, we obtain the existence of a non-normal number $x$ such that $e^{\alpha x}$ is non-normal for every non-zero algebraic $\alpha$.