Exceptional sets of the Oppenheim expansions over the field of formal Laurent series

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• 作者：Mei-Ying Lü ; ^a ; ^{lmy19831102@hotmail.com" class="auth_mail" title="E-mail the corresponding author} ; Jia Liu^b ; ^{liujia860319@163.com" class="auth_mail" title="E-mail the corresponding author} ; Zhen-Liang Zhang^c ; ^{zhliang_zhang@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11K55 ; 11T06 ; 28A80
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：42
• 期：Complete
• 页码：253-268
• 全文大小：329 K

文摘

Let dede6fca9e6b4d04c648d42d5ac36ec5" title="Click to view the MathML source">F_q${\mathbb{F}}_q$ be a finite field with q   elements, F_q((z⁻¹))${\mathbb{F}}_q((z^{-1}))$ denote the field of all formal Laurent series with coefficients in dede6fca9e6b4d04c648d42d5ac36ec5" title="Click to view the MathML source">F_q${\mathbb{F}}_q$ and I   be the valuation ideal of F_q((z⁻¹))${\mathbb{F}}_q((z^{-1}))$. For any formal Laurent series $x={\sum }_{n=\nu }^{\infty }{c}_n{z}^{-n}\in I$, the series $\frac{1}{a_1(x)}+{\sum }_{n=1}^{\infty }\frac{r_1(a_1(x))\cdots r_n(a_n(x))}{s_1(a_1(x))\cdots s_n(a_n(x))}\frac{1}{a_{n+1}(x)}$ is the Oppenheim expansion of x  . Suppose ϕ:N→R⁺$\varphi :\mathbb{N}\to {\mathbb{R}}^{+}$ is a function satisfying ϕ(n)/n→∞$\varphi (n)/n\to \infty$ as n→∞$n\to \infty$. In this paper, we quantify the size, in the sense of Hausdorff dimension, of the set where edae14b123" title="Click to view the MathML source">Δ₀(x)=deg⁡a₁(x)${\mathrm{\Delta }}_0(x)=\mathrm{deg}{a}_1(x)$ and Δ_n(x)=deg⁡a_n+1(x)−2deg⁡a_n(x)−deg⁡r_n(a_n(x))+deg⁡s_n(a_n(x))${\mathrm{\Delta }}_n(x)=\mathrm{deg}{a}_{n+1}(x)-2\mathrm{deg}{a}_n(x)-\mathrm{deg}{r}_n(a_n(x))+\mathrm{deg}{s}_n(a_n(x))$ for all n≥1$n\ge 1$. As applications, we investigate the cases when ϕ(n)$\varphi (n)$ are the given polynomial or exponential functions. At the end of the article, we list some special cases (including Lüroth, Engel, Sylvester expansions of Laurent series and Cantor infinite products of Laurent series) to which we apply the conclusions above.