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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si1.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=dede6fca9e6b4d04c648d42d5ac36ec5" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$ be a finite field with q   elements, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si2.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=f8dd4814ceebe59a366c3c1c1ab96cab" title="Click to view the MathML source">F_q((z⁻¹))class="mathContainer hidden">class="mathCode">${\mathbb{F}}_q((z^{-1}))$ denote the field of all formal Laurent series with coefficients in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si1.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=dede6fca9e6b4d04c648d42d5ac36ec5" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$ and I   be the valuation ideal of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si2.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=f8dd4814ceebe59a366c3c1c1ab96cab" title="Click to view the MathML source">F_q((z⁻¹))class="mathContainer hidden">class="mathCode">${\mathbb{F}}_q((z^{-1}))$. For any formal Laurent series class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si4.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=50efeb6ccd3b07ea1b1383483a357ae6">class="imgLazyJSB inlineImage" height="19" width="149" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1071579716300521-si4.gif">class="mathContainer hidden">class="mathCode">$x={\sum }_{n=\nu }^{\infty }{c}_n{z}^{-n}\in I$, the series class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si5.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=73e8bd0852cb85b5a19b9e8d3d73d05e">class="imgLazyJSB inlineImage" height="25" width="273" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1071579716300521-si5.gif">class="mathContainer hidden">class="mathCode">$\frac{1}{a_1(x)}+{\sum }_{n=1}^{\infty }\frac{r_1(a_1(x))\cdots r_n(a_n(x))}{s_1(a_1(x))\cdots s_n(a_n(x))}\frac{1}{a_{n+1}(x)}$ is the Oppenheim expansion of x  . Suppose class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si239.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=8ffa9e537d5a0fc84c949a092b4810d7" title="Click to view the MathML source">ϕ:N→R⁺class="mathContainer hidden">class="mathCode">$\varphi :\mathbb{N}\to {\mathbb{R}}^{+}$ is a function satisfying class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si7.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=332dc2bad423076d58f02f08fb957b1c" title="Click to view the MathML source">ϕ(n)/n→∞class="mathContainer hidden">class="mathCode">$\varphi (n)/n\to \infty$ as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si241.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=1884890f48e15b6737abbb36a75d00ae" title="Click to view the MathML source">n→∞class="mathContainer hidden">class="mathCode">$n\to \infty$. In this paper, we quantify the size, in the sense of Hausdorff dimension, of the set where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si10.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=e972004b4ec4e08ca529adedae14b123" title="Click to view the MathML source">Δ₀(x)=deg⁡a₁(x)class="mathContainer hidden">class="mathCode">${\mathrm{\Delta }}_0(x)=\mathrm{deg}{a}_1(x)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si11.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=490060a7fdfb1bd1af8c7a8a9155d4bb" title="Click to view the MathML source">Δ_n(x)=deg⁡a_n+1(x)−2deg⁡a_n(x)−deg⁡r_n(a_n(x))+deg⁡s_n(a_n(x))class="mathContainer hidden">class="mathCode">${\mathrm{\Delta }}_n(x)=\mathrm{deg}{a}_{n+1}(x)-2\mathrm{deg}{a}_n(x)-\mathrm{deg}{r}_n(a_n(x))+\mathrm{deg}{s}_n(a_n(x))$ for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si12.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=2a00a9ba181d15b1bd369f5a37049a8f" title="Click to view the MathML source">n≥1class="mathContainer hidden">class="mathCode">$n\ge 1$. As applications, we investigate the cases when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1071579716300521&_mathId=si13.gif&_user=111111111&_pii=S1071579716300521&_rdoc=1&_issn=10715797&md5=f21a3d2c8066f50fe905406d93db9af4" title="Click to view the MathML source">ϕ(n)class="mathContainer hidden">class="mathCode">$\varphi (n)$ 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.