Trace of products in finite fields

Let p be a prime number and let q = p(r). If C and D are large subsets of F-q(*) we study the trace of products cd with c is an element of C and d is an element of D and show that it is well distributed in F-p. We give an optimal condition (up to an absolute constant factor) on the size of the subsets C and D to ensure that the trace of products cd takes any given value in Fp. We also give a condition (optimal up to an absolute constant factor in most cases) on the size of the subsets C and D to ensure that the trace of cd meets the set of k-th powers for k >= 1, respectively the set of generators. Our method will enable us to take sets C and D whose size is substantially below root q. Character sums and Gaussian sums over F-p and F-q will play an important role in the proofs. Some estimates lead to interesting combinatorial questions in finite fields. (C) 2018 Elsevier Inc. All rights reserved.