SOME FRACTALS RELATED TO PARTIAL MAXIMAL DIGITS IN LÜROTH EXPANSION

Fractals Pub Date : 2024-06-04 DOI:10.1142/s0218348x24500786

JIANG DENG, JIHUA MA, KUNKUN SONG, ZHONGQUAN XIE

引用次数: 0

Abstract

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 $α \geq 0$ , the level set $\{x \in (0, 1] : lim_{n \to \infty} \frac{L_{n} (x) log log n}{n} = α \}$ has Hausdorff dimension one. Certain sets of points for which the sequence ${L_{n} (x)}_{n \geq 1}$ grows more rapidly are also investigated.

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与吕洛特展开中部分最大位数有关的一些分形

设[d1(x),d2(x),...,dn(x),...]为 x∈(0,1]的吕洛斯展开，设 Ln(x)=max{d1(x),...dn(x)} 。研究表明，对于任意 α≥0 的水平集 x∈(0,1]:limn→∞Ln(x)loglognn=α，其 Hausdorff 维数为一。还研究了序列 {Ln(x)}n≥1 增长更快的某些点集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Fractals

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