Let $[d_1(x),d_2(x),\dots ,d_n(x),\dots ]$ be the Lüroth expansion of $x\in (0,1]$, and let $L_n(x)=max\{d_1(x),\dots ,d_n(x)\}$. It is shown that for any $\alpha \ge 0$, the level set $$\left\{\right.x\in (0,1]:\underset{n\to \infty }{lim}\frac{L_n(x)loglogn}{n}=\alpha \left\}\right.$$ has Hausdorff dimension one. Certain sets of points for which the sequence ${\{L_n(x)\}}_{n\ge 1}$ grows more rapidly are also investigated.

中文翻译：

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LÜROTH展开式中与部分最大位数相关的一些分形

令 $[d_1(x),d_2(x),\dots ,d_n(x),\dots ]$ 为 $x\in (0,1]$ 的 Lüroth 展开，并令 $L_n(x)=max\{d_1(x),\dots ,d_n(x)\}$ 。结果表明，对于任何 $\alpha \ge 0$ ，水平集 $$\left\{\right.x\in (0,1]:\underset{n\to \infty }{lim}\frac{L_n(x)loglogn}{n}=\alpha \left\}\right.$$ 的 Hausdorff 维数为一。还研究了序列 ${\{L_n(x)\}}_{n\ge 1}$ 增长更快的某些点集。