On the maximal run-length function in the Lüroth expansion
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911483927986176 |
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| author | Yu, Dingding |
| author_facet | Yu, Dingding |
| contents | 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 \( 0 \le α\le β\le 1 \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03889 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the maximal run-length function in the Lüroth expansion Yu, Dingding Metric Geometry Number Theory 11K50 28A80 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 \( 0 \le α\le β\le 1 \). |
| title | On the maximal run-length function in the Lüroth expansion |
| topic | Metric Geometry Number Theory 11K50 28A80 |
| url | https://arxiv.org/abs/2603.03889 |