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Restricted slowly growing digits for infinite iterated function systems

Gerardo González Robert, Mumtaz Hussain, Nikita Shulga, Hiroki Takahasi

Abstract

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $Λ(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in Λ(\mathbf{f}): a_n(x)\in D \text{ for all }n\in\mathbb N \text{ and }\lim_{n\to\infty} a_n=\infty\}. \] For a function $\varphi:\mathbb{N}\to [\min D, \infty)$ such that $\varphi(n)\to\infty$ as $n\to\infty$, we compute the Hausdorff dimension of the set $$ S(\mathbf{f},D,\varphi) = \left\{ x\in \E(\mathbf{f},D) : a_n(x)\leq \varphi(n) \text{ for all } n\in\mathbb N \right\}. $$ We prove that the Hausdorff dimension stays the same no matter how slowly the function $\varphi$ grows. One of the consequences of our result is the recent work of Takahasi (2023), which only dealt with regular continued fraction expansions. We further extend our result to slowly growing products of (not necessarily consecutive) digits.

Restricted slowly growing digits for infinite iterated function systems

Abstract

For an infinite iterated function system on with an attractor and for an infinite subset , consider the set For a function such that as , we compute the Hausdorff dimension of the set We prove that the Hausdorff dimension stays the same no matter how slowly the function grows. One of the consequences of our result is the recent work of Takahasi (2023), which only dealt with regular continued fraction expansions. We further extend our result to slowly growing products of (not necessarily consecutive) digits.
Paper Structure (12 sections, 4 theorems, 64 equations)

This paper contains 12 sections, 4 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Infinite Iterated Function Systems
  3. Statements of results
  4. Examples of infinite iterated function systems
  5. $2$-decaying systems
  6. $d$-decaying systems for $d\neq2$
  7. $\boldsymbol{\xi}$-regular iIFSs, which are not $d$-decaying for any $d$.
  8. Outline of proofs of the theorems
  9. Lower bound of Hausdorff dimension of certain Cantor subsets
  10. Proofs of the main results
  11. Proof of Theorem \ref{['thm:onedigit']}
  12. Proof of Theorem \ref{['thm:product']}

Key Result

Theorem 1.1

Let $D\subseteq \mathbb N$ be an infinite subset and $\varphi\colon\mathbb N\to [\min D,\infty)$ such that $\varphi(n)\to\infty$ as $n \to\infty$. Then,

Theorems & Definitions (5)

  • Theorem 1.1: MR4607611, Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof