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

In this paper, we use methods from spectral graph theory to obtain some results on the sum-product problem over finite valuation rings $\mathcal{R}$ of order $q^r$ which generalize recent results given by Hegyv'ari and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$, if $A$ and $B$ are two sets in ${\mathcal{R}}^{\ast }$ with $\mathrm{\mid }A\mathrm{\mid }=\mathrm{\mid }B\mathrm{\mid }=q^{\alpha }$, then $$\max \{\mathrm{\mid }f(A,B)\mathrm{\mid },\mathrm{\mid }g(A,B)\mathrm{\mid }\}\gtrsim \mathrm{\mid }A{\mathrm{\mid }}^{1+\mathrm{\Delta }(\alpha )},$$ for some $\mathrm{\Delta }(\alpha )>0$.