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No Nowhere Constant Continuous Function Maps All Non-Normal Numbers to Normal Numbers

Chokri Manai

TL;DR

The paper proves that there is no nowhere constant continuous function on an open interval that maps all non-normal numbers into normal numbers, revealing the intricate topological structure of the non-normal set $\mathcal{N}^c$ via the notion of super-density. It constructs a zero-Hausdorff-dimension, super-dense subset $\mathcal{Z}_b$ of $\mathcal{N}_{b,s}^{c}$ and a Cantor-type enrichment $\hat{C}$ that sends normals to non-normals, highlighting a stark asymmetry between $\mathcal{N}$ and $\mathcal{N}^c$. The work connects these phenomena to Baire category, showing that super-dense sets must be non-meager and that co-meager $G_{\delta}$ sets are super-dense, while also providing non-constructive examples where both a set and its complement are super-dense. The results illuminate the geometric richness of non-normal numbers and offer constructive avenues for generating simultaneous non-normality, with potential implications for understanding the distribution of normal numbers and related dynamical/number-theoretic structures.

Abstract

In this work, we consider the set of non-normal numbers $\mathcal{N}^c$ and ask if there is a non-empty open interval $I$ and a nowhere constant continuous function $\varphi : I \to \mathbb{R}$ which maps all non-normal numbers to normal numbers, i.e., $\varphi(I \cap \mathcal{N}^c) \subset \mathcal{N}.$ We answer this question negatively. We call a set with this property \textit{super-dense}. The super-density of $\mathcal{N}^c$ illustrates further its topological size or geometric richness. Surprisingly, the measure-theoretically ``larger" set of normal numbers $\mathcal{N}$ does not share this property and we give an explicit Cantor-type construction of a nowhere constant continuous function $\hat{C}$, which maps all normal numbers to non-normal numbers. Our final result discusses the relation between super-density and the Baire categories employing various set-theoretic strategies.

No Nowhere Constant Continuous Function Maps All Non-Normal Numbers to Normal Numbers

TL;DR

The paper proves that there is no nowhere constant continuous function on an open interval that maps all non-normal numbers into normal numbers, revealing the intricate topological structure of the non-normal set via the notion of super-density. It constructs a zero-Hausdorff-dimension, super-dense subset of and a Cantor-type enrichment that sends normals to non-normals, highlighting a stark asymmetry between and . The work connects these phenomena to Baire category, showing that super-dense sets must be non-meager and that co-meager sets are super-dense, while also providing non-constructive examples where both a set and its complement are super-dense. The results illuminate the geometric richness of non-normal numbers and offer constructive avenues for generating simultaneous non-normality, with potential implications for understanding the distribution of normal numbers and related dynamical/number-theoretic structures.

Abstract

In this work, we consider the set of non-normal numbers and ask if there is a non-empty open interval and a nowhere constant continuous function which maps all non-normal numbers to normal numbers, i.e., We answer this question negatively. We call a set with this property \textit{super-dense}. The super-density of illustrates further its topological size or geometric richness. Surprisingly, the measure-theoretically ``larger" set of normal numbers does not share this property and we give an explicit Cantor-type construction of a nowhere constant continuous function , which maps all normal numbers to non-normal numbers. Our final result discusses the relation between super-density and the Baire categories employing various set-theoretic strategies.

Paper Structure

This paper contains 6 sections, 14 theorems, 63 equations.

Table of Contents

  1. Introduction and Main Result
  2. Proof of Main Results
  3. Proof of Theorem \ref{['thm:main2']}
  4. Proof of Theorem \ref{['thm:topology']}
  5. Proof of Theorem \ref{['thm:main']}
  6. Proof of Lemma \ref{['lem:dim']}

Key Result

Theorem 1.2

Let $I \subset {\mathbb{R}}$ be any non-empty open interval and $\varphi \colon I \to {\mathbb{R}}$ a nowhere constant continuous function. Then, for every base $b \geq 2$ In fact, we have more generally for every countable collection $(\varphi_k)_{k \in {\mathbb{N}}}$ of nowhere continuous functions $\varphi_k \colon I \to {\mathbb{R}}$,

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 17 more