New estimates for exponential sums over multiplicative subgroups and intervals in prime fields

Abstract

Let $\mathcal{H}$ be a multiplicative subgroup of ${\mathbb{F}}_p^{\ast }$ of order $H>p^{1/4}$. We show that $$\underset{(a,p)=1}{\max }\left|\sum _{x\in \mathcal{H}}{\mathbf{e}}_p(ax)\right|\le H^{1-31/2880+o(1)},$$ where ${\mathbf{e}}_p(z)=\exp (2\pi iz/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product $nx$ with $x\in \mathcal{H}$ and $n\in \mathcal{N}$ for a short interval $\mathcal{N}$ of consecutive integers.

14 pages