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 $$\sum _{x\in H}\chi (x+a),a\in {\mathbb{F}}_p^{\ast },$$ where $H<{\mathbb{F}}_p^{\ast }$ is a multiplicative subgroup in finite prime field ${\mathbb{F}}_p$. Some interesting mean-value estimates are also provided.

arXiv comment7 pages