Conditional expanding bounds for two-variable functions over arbitrary fields

In this paper, we prove some results on the sum-product problem over arbitrary fields which improve and generalize results given by Hegyvári and Hennecart [5]. 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 an arbitrary field F with |A|=|B|, thenmax⁡{|f(A,B)|,|g(A,B)|}≫|A|1+c, for some c>0.