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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.04159 |
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| _version_ | 1866910461550657536 |
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| author | Li, Zhihui Liao, Xin Yu, Dingding |
| author_facet | Li, Zhihui Liao, Xin Yu, Dingding |
| contents | Each $x\in (0,1]$ can be uniquely expanded as a power-2-decaying Gauss-like expansion, in the form of \begin{equation*} x=\sum_{i=1}^{\infty}2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))},\qquad d_i(x)\in \mathbb{N}. \end{equation*} Let $ϕ:\mathbb{N}\to \mathbb{R}^{+}$ be an arbitrary positive function. We are interested in the size of the set $$F(ϕ)=\{x\in (0,1]:d_n(x)\ge ϕ(n)~~\text{for infinity many}~n\}.$$ We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of $F(ϕ)$. When the Lebesgue measure of $F(ϕ)$ is zero, we calculate its Hausdorff dimension. Furthermore, we analyse the growth rate of the maximal digit among the first $n$ digits from probability and multifractal perspectives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04159 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metrical theory of power-2-decaying Gauss-like expansion Li, Zhihui Liao, Xin Yu, Dingding Number Theory Each $x\in (0,1]$ can be uniquely expanded as a power-2-decaying Gauss-like expansion, in the form of \begin{equation*} x=\sum_{i=1}^{\infty}2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))},\qquad d_i(x)\in \mathbb{N}. \end{equation*} Let $ϕ:\mathbb{N}\to \mathbb{R}^{+}$ be an arbitrary positive function. We are interested in the size of the set $$F(ϕ)=\{x\in (0,1]:d_n(x)\ge ϕ(n)~~\text{for infinity many}~n\}.$$ We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of $F(ϕ)$. When the Lebesgue measure of $F(ϕ)$ is zero, we calculate its Hausdorff dimension. Furthermore, we analyse the growth rate of the maximal digit among the first $n$ digits from probability and multifractal perspectives. |
| title | Metrical theory of power-2-decaying Gauss-like expansion |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.04159 |