An elementary approach to character sums over multiplicative subgroups

Abstract: Let $p$ be an odd prime. Using I. M. Vinogradov's bilinear estimate, we present an elementary approach to estimate nontrivially the character sum $$ \sum_{x\in H}\chi(x+a),\qquad a\in\Bbb F_p^*, $$ where $H<\Bbb F_p^*$ is a multiplicative subgroup in finite prime field $\Bbb F_p$. Some interesting mean-value estimates are also provided.