On the exponential sum—product problem

Let g be an element of order T   over a finite field FpFp of p elements, where p is a prime. We show that for a very wide class of sets A, B ∈ {1, . . . , T} at least one of the sets {gab:a∈A,b∈B}and{ga+gb:a∈A,b∈B}{gab:a∈A,b∈B}and{ga+gb:a∈A,b∈B} is large.