Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the maximal run-length function in the Lüroth expansion

Dingding Yu

Abstract

Let \( \ell_n(x) \) denote the maximal run-length among the first \( n \) digits of the Lüroth expansion of \( x\in(0,1] \). While \( \ell_n(x) \) grows logarithmically, we investigate the finer multifractal properties of the exceptional set where $\ell_n(x)$ exhibits linear growth. Specifically, we establish the Hausdorff dimension of the set \[ \left\{ x \in (0,1] : \liminf_{n \to \infty} \frac{\ell_n(x)}{n} = α, \; \limsup_{n \to \infty} \frac{\ell_n(x)}{n} = β\right\}, \] for all \( 0 \le α\le β\le 1 \).

On the maximal run-length function in the Lüroth expansion

Abstract

Let \( \ell_n(x) \) denote the maximal run-length among the first digits of the Lüroth expansion of . While \( \ell_n(x) \) grows logarithmically, we investigate the finer multifractal properties of the exceptional set where exhibits linear growth. Specifically, we establish the Hausdorff dimension of the set \[ \left\{ x \in (0,1] : \liminf_{n \to \infty} \frac{\ell_n(x)}{n} = α, \; \limsup_{n \to \infty} \frac{\ell_n(x)}{n} = β\right\}, \] for all .
Paper Structure (14 sections, 15 theorems, 99 equations)

This paper contains 14 sections, 15 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['mainthm']}: Upper Bound
  4. Case 1: $\beta = 0$
  5. Case 2: $\beta = 1$
  6. Case 3: $0 < \beta < 1$
  7. Case 3.1: $\alpha = 0$
  8. Case 3.2: $0 < \alpha \le \beta<1$
  9. Proof of Theorem \ref{['mainthm']}: Lower bound
  10. Construct subset of $E(\alpha,\beta)$
  11. Symbolic description of $f(G(M))$
  12. Measure supported on $f(G_{(\{\widetilde{n_k}\},\{m_k\})}(M))$
  13. Gap between fundamental intervals
  14. Completion of the lower bound

Key Result

Theorem 1.1

sun2018maximal For almost all $x\in(0,1]$, $\lim\limits_{n\to\infty}\ell_n(x)/\log_2 n=1$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • ...and 12 more