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 q(r) which generalize recent results given by Hegyvari 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 vertical bar A vertical bar = vertical bar B vertical bar = q(alpha), then max {vertical bar f(A, B)vertical bar, vertical bar g(A, B)vertical bar} >> vertical bar A vertical bar(1+Delta(alpha)), for some Delta(alpha) > 0. (C) 2016 Elsevier Ltd. All rights reserved.