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- W4380558575 abstract "The Luroth expansion of a real number $xin (0,1]$ is the series [ x= frac{1}{d_1} + frac{1}{d_1(d_1-1)d_2} + frac{1}{d_1(d_1-1)d_2(d_2-1)d_3} + cdots, ] with $d_jinmathbb{N}_{geq 2}$ for all $jinmathbb{N}$. Given $min mathbb{N}$, $mathbf{t}=(t_0,ldots, t_{m-1})inmathbb{R}_{>0}^{m-1}$ and any function $Psi:mathbb{N}to (1,infty)$, define [ mathcal{E}_{mathbf{t}}(Psi)colon= left{ xin (0,1]: d_n^{t_0} cdots d_{n+m}^{t_{m-1}}geq Psi(n) text{ for infinitely many} n inmathbb{N} right}. ] We establish a Lebesgue measure dichotomy statement (a zero-one law) for $mathcal{E}_{mathbf{t}}(Psi)$ under a natural non-removable condition $liminf_{ntoinfty} Psi(n)>~1$. Let $B$ be given by [ log B colon= liminf_{ntoinfty} frac{log(Psi(n))}{n}. ] For any $minmathbb{N}$, we compute the Hausdorff dimension of $mathcal{E}_{mathbf{t}}(Psi)$ when either $B=1$ or $B=infty$. We also compute the Hausdorff dimension of $mathcal{E}_{mathbf{t}}(Psi)$ when $1<B< infty$ for $m=2$." @default.
- W4380558575 created "2023-06-14" @default.
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- W4380558575 date "2023-06-12" @default.
- W4380558575 modified "2023-09-27" @default.
- W4380558575 title "Metrical properties of weighted products of consecutive Luroth digits" @default.
- W4380558575 doi "https://doi.org/10.48550/arxiv.2306.06886" @default.
- W4380558575 hasPublicationYear "2023" @default.
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