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 R of order qr 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 R∗ with |A|=|B|=qα, then max{|f(A,B)|,|g(A,B)|}≫|A|1+Δ(α), for some Δ(α)>0.