Trace of products in finite fields

Let p be a prime number and let q=pr. If C and D are large subsets of Fq⁎ we study the trace of products cd with c∈C and d∈D and show that it is well distributed in Fp. 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 q. Character sums and Gaussian sums over Fp and Fq will play an important role in the proofs. Some estimates lead to interesting combinatorial questions in finite fields.