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 Hegyvári 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 \(\Delta(\alpha)>0\).