Double Character Sums over Subgroups and Intervals

We estimate double sums $$S_\chi(a, I, G) = \sum_{x \in I} \sum_{\lambda \in G} \chi(x + a\lambda), \qquad 1\le a < p-1,$$ with a multiplicative character $\chi$ modulo $p$ where $I= \{1,\ldots, H\}$ and $G$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_\chi(a, I, G)$ can be derived from the Burgess bound if $H \ge p^{1/4+\varepsilon}$ and from some standard elementary arguments if $T \ge p^{1/2+\varepsilon}$, where $\varepsilon>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$T_\chi(a, G) = \sum_{\lambda, \mu \in G} \chi(a + \lambda + \mu), \qquad 1\le a < p-1,$$ and give an application to primitive roots modulo $p$ with $3$ non-zero binary digits.