Double character sums over subgroups and intervals

We estimate double sums S_χ(a, I, G) = ∑_x∈I∑_λ∈Gχ(x + aλ), 1 ≤ a < p - 1, with a multiplicative character χ modulo p where I = {1, . . . , H} and G is a subgroup of order T of the multiplicative group of the finite field of p elements. A nontrivial upper bound on S _χ (a, I, G) can be derived from the Burgess bound if H ≥ p^1/4+ε and from some standard elementary arguments if T ≥ p^1/2+ε, where ε > 0 is arbitrary. We obtain a nontrivial estimate in a wider range of parameters H and T . We also estimate double sums T_χ(a, G) = ∑_χ,μ∈G χ(a + λ + μ), 1 ≤ a < p - 1, and give an application to primitive roots modulo p with three nonzero binary digits.