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It is well known that every $x\in(0,1]$ can be expanded to an infinite L\"{u}roth series with the form of \[ x=\tfrac{1}{d_1(x)}+\cdots+\tfrac{1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}+\cdots, \] where $d_n(x)\geq 2$ for all $n\geq 1$. In this paper, the set of points with some restrictions on the digits in L\"{u}roth series expansions are considered. Namely, the Hausdorff dimension of following the set $$ F_{\phi}=\{x\in (0,1]: d_n(x)\geq \phi(n), \ {\rm{i. \ o.}} \ n\} $$ is determined, where $\phi$ is an integer-valued function defined on $\mathbb{N}$, and $\phi(n)\to \infty$ as $n\to \infty$.
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참고문헌 (17)
참고문헌 신청
L. Barreira / 2009 / Frequency of digits in the Luroth expansion / J. Number Theory 129(6):1479~1490
K. Dajani / 2002 / Ergodic Theory of Numbers / Mathematical Association of America
K. Dajani / 1996 / On approximation by Luroth series / J. Theor. Nombres Bordeaux 8(2):331~346
K. J. Falconer / 1990 / Fractal Geometry: Mathematical Foundations and Application / Wiley
K. J. Falconer / 1997 / Techniques in Fractal Geometry / Wiley
K. Dajani / 2002 / Ergodic Theory of Numbers / Mathematical Association of America
K. Dajani / 1996 / On approximation by Luroth series / J. Theor. Nombres Bordeaux 8(2):331~346
K. J. Falconer / 1990 / Fractal Geometry: Mathematical Foundations and Application / Wiley
K. J. Falconer / 1997 / Techniques in Fractal Geometry / Wiley
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