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
46ae16d28d269" title="Click to view the MathML source">f(x,y) and
g(x,y), if
A and
a61114667f2672380f808f5e52" title="Click to view the MathML source">B are two sets in
R∗ with
|A|=|B|=qα, then
for some
Δ(α)>0.