On the maximal run-length function in the L\"uroth expansion

TL;DR

This paper studies the multifractal properties of the set of points in (0,1] where the maximal run-length in the L"uroth expansion grows linearly, establishing their Hausdorff dimension for various growth rates.

Contribution

It provides a detailed analysis of the Hausdorff dimension of sets characterized by linear growth rates of maximal run-length in L"uroth expansions, revealing their multifractal structure.

Findings

Hausdorff dimension of sets with specified linear growth rates is established.

Multifractal spectrum of the maximal run-length function is characterized.

Results extend understanding of digit pattern complexity in L"uroth expansions.

Abstract

Let \( \ell_n(x) \) denote the maximal run-length among the first \( n \) digits of the L\"{u}roth 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 \[ \left\{ x \in (0,1] : \liminf_{n \to \infty} \frac{\ell_n(x)}{n} = \alpha, \; \limsup_{n \to \infty} \frac{\ell_n(x)}{n} = \beta \right\}, \] for all \( 0 \le \alpha \le \beta \le 1 \).