A note on Engel series expansions of Laurent series and Hausdorff dimensions

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• 作者：Meiying Lü ; ^{lmy19831102@hotmail.com" class="auth_mail}
• 关键词：11K55 ; 11T06 ; 28A80
• 刊名：Journal of Number Theory
• 出版年：September 2014
• 年：2014
• 卷：142
• 期：Complete
• 页码：44-50
• 全文大小：228 K

文摘

Let F_q${\mathbb{F}}_q$ be the finite field with q   elements and F_q((z⁻¹))${\mathbb{F}}_q((z^{-1}))$ be the field of all formal Laurent series with coefficients in F_q${\mathbb{F}}_q$. For any x∈I:=z⁻¹F_q((z⁻¹))$x\in I:=z^{-1}{\mathbb{F}}_q((z^{-1}))$, the Engel series expansion of x   is with a_j(x)∈F_q[z]$a_j(x)\in {\mathbb{F}}_q[z]$. Suppose that 蠒:N→R⁺$蠒:\mathbb{N}\to {\mathbb{R}}^{+}$ is a function satisfying 蠒(n)猢緉$蠒(n)猢?/mo><mi>n</mi>$ for all integers n   large enough. In this note, we consider the following set and establish a lower bound of its Hausdorff dimension. As a direct application, we obtain in particular ${\dim }_{\mathrm{H}}\{x\in I:{\lim }_{n\to \infty }\frac{\mathrm{deg}{a}_n(x)}{n^{尾}}=纬\}=1$ (where 尾>1$尾>1$, 纬>0$纬>0$ or 尾=1$尾=1$, 纬猢?$纬猢?/mo><mn>1</mn>$, and dim_H${\dim }_{\mathrm{H}}$ denotes the Hausdorff dimension), which generalizes a result of J. Wu dated 2003.