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The Theory of Normality for Dynamically Generated Cantor Series Expansions

Sohail Farhangi, Bill Mance

TL;DR

This work develops a comprehensive framework to study normality for dynamically generated Cantor-series expansions, extending classical base-$g$ theory to the $Q$-Cantor setting. By generating $Q=(q_n)$ from a measure-preserving system via a function $f$, the authors define and relate multiple normality notions ($ ext{N}(Q)$, $ ext{DN}(Q)$, $ ext{UN}(Q)$, $ ext{UDN}(Q)$) and prove that under ergodic, zero-entropy dynamics with finite $\int \log(f)\,dta$, the standard and distributional notions coincide. They also establish Hot Spot theorems for these dynamically generated sequences and specialize to $g$-power sequences to recover classical base-$g$ normality $ ext{N}_g$; a rich set of examples and counterexamples clarifies the necessity of hypotheses. The paper concludes with questions and conjectures about selection rules, invariance properties, and the deeper dynamical structure of the associated skew-product systems, laying groundwork for a broad Cantor-series normality theory with parallels to base expansions.

Abstract

The theory of normality for base $g$ expansions of real numbers in $[0,1)$ is rich and well developed. Similar theories have been developed for many other numeration systems, such as the regular continued fraction expansion, $β$-expansions, and Lüroth series expansions. Let $Q=(q_n)_{n \in \mathbb{N}}$ be a sequence of integers greater than or equal to 2. The $Q$-Cantor series expansion of $x \in [0,1)$ is the unique sum of the form $x=\sum_{n=1}^\infty \frac{x_n}{q_1q_2\cdots q_n}$, where $x_n \neq q_n-1$ infinitely often. For the Cantor series expansions, most of the literature thus far considers $Q$ where the theory of normality differs drastically from that of the base $g$ expansions. We introduce the class of dynamically generated Cantor series expansions, which is a large class of Cantor series expansions for which much of the classical theory of base $g$ expansions can be developed in parallel. This class includes many examples such as the Thue-Morse sequence on $\{2,3\}$ and translated Champernowne numbers. A special case of our main results is that if $Q$ is a bounded basic sequence that is dynamically generated by an ergodic system having zero entropy, then normality base $Q$ coincides with distribution normality base $Q$, and $Q$ possesses a Hot Spot Theorem.

The Theory of Normality for Dynamically Generated Cantor Series Expansions

TL;DR

This work develops a comprehensive framework to study normality for dynamically generated Cantor-series expansions, extending classical base- theory to the -Cantor setting. By generating from a measure-preserving system via a function , the authors define and relate multiple normality notions (, , , ) and prove that under ergodic, zero-entropy dynamics with finite , the standard and distributional notions coincide. They also establish Hot Spot theorems for these dynamically generated sequences and specialize to -power sequences to recover classical base- normality ; a rich set of examples and counterexamples clarifies the necessity of hypotheses. The paper concludes with questions and conjectures about selection rules, invariance properties, and the deeper dynamical structure of the associated skew-product systems, laying groundwork for a broad Cantor-series normality theory with parallels to base expansions.

Abstract

The theory of normality for base expansions of real numbers in is rich and well developed. Similar theories have been developed for many other numeration systems, such as the regular continued fraction expansion, -expansions, and Lüroth series expansions. Let be a sequence of integers greater than or equal to 2. The -Cantor series expansion of is the unique sum of the form , where infinitely often. For the Cantor series expansions, most of the literature thus far considers where the theory of normality differs drastically from that of the base expansions. We introduce the class of dynamically generated Cantor series expansions, which is a large class of Cantor series expansions for which much of the classical theory of base expansions can be developed in parallel. This class includes many examples such as the Thue-Morse sequence on and translated Champernowne numbers. A special case of our main results is that if is a bounded basic sequence that is dynamically generated by an ergodic system having zero entropy, then normality base coincides with distribution normality base , and possesses a Hot Spot Theorem.

Paper Structure

This paper contains 11 sections, 22 theorems, 120 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Normal numbers
  3. Dynamically generated basic sequences
  4. Main Results
  5. Examples
  6. Counterexamples
  7. Properties of Dynamically Generated Basic Sequences
  8. Hot Spot Theorems and equivalent notions of normality
  9. Ratio Normality
  10. $g$-power basic sequences
  11. Questions and Conjectures

Key Result

Theorem 1.1

A real number $x$ is normal in base $g$ if and only if the sequence $(g^nx)_{n = 1}^\infty$ is uniformly distributed mod $1$.

Figures (4)

  • Figure 1: $E_B$ and $E_B\times I_{D,B}$ when $\ell = 1$.
  • Figure 2: $E_B$ and $E_B\times I_{D,B}$ when $\ell = 2$.
  • Figure 3: $E_B$ and $E_B\times I_{D,B}$ when $\ell = 3$.
  • Figure 4: $E_B$ and $E_B\times I_{D,B}$ for selected values of $D$ when $\ell = 3$.

Theorems & Definitions (67)

  • Theorem 1.1: D. D. Wall
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: Bergelson-Vandehey HotSpotProofOfGeneralizedWallTheorem
  • Theorem 1.9: Hot Spot
  • Example 2.1: Periodic sequences
  • Example 2.2: Almost Periodic sequences
  • ...and 57 more