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Finite pattern problems related to Engel expansion

Chun-Yun Cao, Yang Xiao

Abstract

Let $\mathcal{F}$ be a countable collection of functions $f$ defined on the integers with integer values, such that for every $f\in \mathcal{F}$, $f(n)\to +\infty$ as $n\to +\infty$. This paper primarily investigates the Hausdorff dimension of the set of points whose digit sequences of the Engel expansion are strictly increasing and contain any finite pattern of $\mathcal{F}$, demonstrating applications with representative examples.

Finite pattern problems related to Engel expansion

Abstract

Let be a countable collection of functions defined on the integers with integer values, such that for every , as . This paper primarily investigates the Hausdorff dimension of the set of points whose digit sequences of the Engel expansion are strictly increasing and contain any finite pattern of , demonstrating applications with representative examples.

Paper Structure

This paper contains 6 sections, 12 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem 1.2
  4. Subset of $E_{\mathcal{F}}$
  5. Hausdorff dimension of $E_{\mathcal{F}}$
  6. Applications of Theorem\ref{['thF']}

Key Result

Theorem 1.1

For any set $\mathcal{F}$ of countable many functions satisfying f, $\dim_HE_{\mathcal{F}}=1$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Proposition 2.2
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.7
  • Lemma 3.1
  • ...and 3 more