Metric properties of continued fractions with large prime partial quotients
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913131306942464 |
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| author | Cheng, Wanjin Wu, Wen |
| author_facet | Cheng, Wanjin Wu, Wen |
| contents | Let $x \in [0,1)$ with continued fraction expansion $[a_1(x),a_2(x),\dots]$, and let $ϕ:\mathbb{N}\to\mathbb{R}^+$ be a non-decreasing function. We consider the numbers whose continued fraction expansions contain at least two partial quotients that are simultaneously large and prime, that is \[ E'(ϕ):=\Big\{x\in[0,1): \exists\, 1\leq k\neq l\leq n, \ a'_{k}(x),\ a'_{l}(x)\geqϕ(n) \ \text{for i.m. } n\in\mathbb{N}\Big\}, \] where $a'_i(x)$ denotes $a_i(x)$ if $a_i(x)$ is prime and $0$ otherwise. We establish a zero-one law for the Lebesgue measure of $E'(ϕ)$ and determine its Hausdorff dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_27284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric properties of continued fractions with large prime partial quotients Cheng, Wanjin Wu, Wen Number Theory Primary 11K50, Secondary 28A80, 11J83 Let $x \in [0,1)$ with continued fraction expansion $[a_1(x),a_2(x),\dots]$, and let $ϕ:\mathbb{N}\to\mathbb{R}^+$ be a non-decreasing function. We consider the numbers whose continued fraction expansions contain at least two partial quotients that are simultaneously large and prime, that is \[ E'(ϕ):=\Big\{x\in[0,1): \exists\, 1\leq k\neq l\leq n, \ a'_{k}(x),\ a'_{l}(x)\geqϕ(n) \ \text{for i.m. } n\in\mathbb{N}\Big\}, \] where $a'_i(x)$ denotes $a_i(x)$ if $a_i(x)$ is prime and $0$ otherwise. We establish a zero-one law for the Lebesgue measure of $E'(ϕ)$ and determine its Hausdorff dimension. |
| title | Metric properties of continued fractions with large prime partial quotients |
| topic | Number Theory Primary 11K50, Secondary 28A80, 11J83 |
| url | https://arxiv.org/abs/2510.27284 |