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 $\mathcal{R}^*$ with $|A|=|B|=q^\alpha$, then $$\max\left\lbrace |f(A, B)|, |g(A, B)| \right\rbrace\gtrsim |A|^{1+\Delta(\alpha)},$$ for some $\Delta(\alpha)>0$.