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arXiv · 2603.03889

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

Dingding Yu
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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 \( 0 \le \alpha \le \beta \le 1 \).

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Dingding Yu. 2026-03-04. On the maximal run-length function in the L\"uroth expansion. https://arxiv.org/abs/2603.03889

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