SUM-PRODUCT PHENOMENON IN FINITE FIELDS NOT OF PRIME ORDER

Let F = F-p(n) be a finite field and A a subset of F so that for any A' subset of A with vertical bar A'vertical bar >= vertical bar A vertical bar(15/16) and for any G subset of F a subfield (not necessarily proper) and for any elements c, d is an element of F if A' subset of cG + d, then vertical bar A vertical bar <= vertical bar G vertical bar(1/2) Then it must be that max(vertical bar A + A vertical bar,vertical bar F(A, A)vertical bar) greater than or similar to vertical bar A vertical bar(17/16) where F : F-p x F-p -> F-p is a function defined by F(x, y) = x(g(x) + cy), where c is an element of F-p*; and g : F-p -> F-p is any function. The case g = 0 and c = 1 improves the exponent in [6] from 20/19 to 17/16.