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 R$\mathcal{R}$ of order q^r$q^r$ which generalize recent results given by Hegyvári and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions f(x,y)$f(x,y)$ and g(x,y)$g(x,y)$, if A$A$ and B$B$ are two sets in R^∗${\mathcal{R}}^{\ast }$ with |A|=|B|=q^α$\left|A\right|=\left|B\right|=q^{\alpha }$, then for some 575460264c0956" title="Click to view the MathML source">Δ(α)>0$\Delta \left(\alpha \right)>0$.