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Double Character Sums over Subgroups and Intervals

We estimate double sums $$ S_χ(a, I, G) = \sum_{x \in I} \sum_{λ\in G} χ(x + aλ), \qquad 1\le a < p-1, $$ with a multiplicative character $χ$ 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 $S_χ(a, I, G)$ 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 $$ T_χ(a, G) = \sum_{λ, μ\in G} χ(a + λ+ μ), \qquad 1\le a < p-1, $$ and give an application to primitive roots modulo $p$ with $3$ non-zero binary digits.