Multifractal analysis of maximal product of consecutive partial quotients in continued fractions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911002483752960 |
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| author | Song, Kunkun Yu, Dingding Yu, Yueli |
| author_facet | Song, Kunkun Yu, Dingding Yu, Yueli |
| contents | Let $[a_1(x), a_2(x), \ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. We study the growth rate of the maximal product of consecutive partial quotients among the first $n$ terms, defined by $L_n(x)=\max_{1\leq i\leq n}\{a_i(x)a_{i+1}(x)\}$, from the viewpoint of multifractal analysis. More precisely, we determine the Hausdorff dimension of the level set \[L(φ):=\left\{x\in (0,1):\lim_{n\to \infty}\frac{L_n(x)}{φ(n)}=1\right\},\] where $φ:\mathbb{R^+}\to\mathbb{R^+}$ is an increasing function such that $\log φ$ is a regularly increasing function with index $ρ$. We show that there exists a jump of the Hausdorff dimension of $L(φ)$ when $ρ=1/2$. We also construct uncountably many discontinuous functions $ψ$ that cause the Hausdorff dimension of $L(ψ)$ to transition continuously from 1 to 1/2, filling the gap when $ρ=1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multifractal analysis of maximal product of consecutive partial quotients in continued fractions Song, Kunkun Yu, Dingding Yu, Yueli Number Theory 11K50, 28A80 Let $[a_1(x), a_2(x), \ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. We study the growth rate of the maximal product of consecutive partial quotients among the first $n$ terms, defined by $L_n(x)=\max_{1\leq i\leq n}\{a_i(x)a_{i+1}(x)\}$, from the viewpoint of multifractal analysis. More precisely, we determine the Hausdorff dimension of the level set \[L(φ):=\left\{x\in (0,1):\lim_{n\to \infty}\frac{L_n(x)}{φ(n)}=1\right\},\] where $φ:\mathbb{R^+}\to\mathbb{R^+}$ is an increasing function such that $\log φ$ is a regularly increasing function with index $ρ$. We show that there exists a jump of the Hausdorff dimension of $L(φ)$ when $ρ=1/2$. We also construct uncountably many discontinuous functions $ψ$ that cause the Hausdorff dimension of $L(ψ)$ to transition continuously from 1 to 1/2, filling the gap when $ρ=1/2$. |
| title | Multifractal analysis of maximal product of consecutive partial quotients in continued fractions |
| topic | Number Theory 11K50, 28A80 |
| url | https://arxiv.org/abs/2502.02064 |