Conditional expanding bounds for two-variable functions over finite valuation rings

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• 作者：Le Quang Ham^a ; ^{hamlaoshi@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Pham Van Thang^b ; ^{thang.pham@epfl.ch" class="auth_mail" title="E-mail the corresponding author} ; Le Anh Vinh^c ; ^{vinhla@vnu.edu.vn" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：60
• 期：Complete
• 页码：114-123
• 全文大小：414 K

文摘

In this paper, we use methods from spectral graph theory to obtain some results on the sum–product problem over finite valuation rings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si1.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=f5b95ec2bf3bc99124783bdd794b41ea" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">$\mathcal{R}$ of order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si2.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=7cfe0f840f38ea17d199b860a602b5af" title="Click to view the MathML source">q^rclass="mathContainer hidden">class="mathCode">$q^r$ which generalize recent results given by Hegyvári and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si3.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=892d3b882569a7461ed46ae16d28d269" title="Click to view the MathML source">f(x,y)class="mathContainer hidden">class="mathCode">$f(x,y)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si4.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=431d1407dfe554e0e1f51bd858e738ea" title="Click to view the MathML source">g(x,y)class="mathContainer hidden">class="mathCode">$g(x,y)$, if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si5.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=4250ed32078001a74daf4e5e07e674d3" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$A$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si6.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=8d077ea61114667f2672380f808f5e52" title="Click to view the MathML source">Bclass="mathContainer hidden">class="mathCode">$B$ are two sets in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si7.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=4c989ef6bfb24c48db4a96699bf01f72" title="Click to view the MathML source">R^∗class="mathContainer hidden">class="mathCode">${\mathcal{R}}^{\ast }$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si8.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=f09c3daae10a0ce8821c9f42d86800f5" title="Click to view the MathML source">|A|=|B|=q^αclass="mathContainer hidden">class="mathCode">$\left|A\right|=\left|B\right|=q^{\alpha }$, then for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300865&_mathId=si10.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=502a6f0ca7c4635d88575460264c0956" title="Click to view the MathML source">Δ(α)>0class="mathContainer hidden">class="mathCode">$\Delta \left(\alpha \right)>0$.