In this paper, we use methods from spectral graph theory to obtain some results on the sum–product problem o
ver finite
valuation rings
ve&_eid=1-s2.0-S0195669816300865&_mathId=si1.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=f5b95ec2bf3bc99124783bdd794b41ea" title="Click to view the MathML source">R of order
ve&_eid=1-s2.0-S0195669816300865&_mathId=si2.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=7cfe0f840f38ea17d199b860a602b5af" title="Click to view the MathML source">qr which generalize recent results gi
ven by Hegy
vári and Hennecart (2013). More precisely, we pro
ve that, for related pairs of two-
variable functions
ve&_eid=1-s2.0-S0195669816300865&_mathId=si3.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=892d3b882569a7461ed46ae16d28d269" title="Click to view the MathML source">f(x,y) and
ve&_eid=1-s2.0-S0195669816300865&_mathId=si4.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=431d1407dfe554e0e1f51bd858e738ea" title="Click to view the MathML source">g(x,y), if
ve&_eid=1-s2.0-S0195669816300865&_mathId=si5.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=4250ed32078001a74daf4e5e07e674d3" title="Click to view the MathML source">A and
ve&_eid=1-s2.0-S0195669816300865&_mathId=si6.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=8d077ea61114667f2672380f808f5e52" title="Click to view the MathML source">B are two sets in
ve&_eid=1-s2.0-S0195669816300865&_mathId=si7.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=4c989ef6bfb24c48db4a96699bf01f72" title="Click to view the MathML source">R∗ with
ve&_eid=1-s2.0-S0195669816300865&_mathId=si8.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=f09c3daae10a0ce8821c9f42d86800f5" title="Click to view the MathML source">|A|=|B|=qα, then
v class="formula" id="fd000005">v class="mathml">ve&_eid=1-s2.0-S0195669816300865&_mathId=si9.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=a6057381ae00e05828007059ade3bf80" title="Click to view the MathML source">max{|f(A,B)|,|g(A,B)|}≫|A|1+Δ(α),v>v> for some
ve&_eid=1-s2.0-S0195669816300865&_mathId=si10.gif&_user=111111111&_pii=S0195669816300865&_rdoc=1&_issn=01956698&md5=502a6f0ca7c4635d88575460264c0956" title="Click to view the MathML source">Δ(α)>0.