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

Daniel di Benedetto, Moubariz Z. Garaev, Víctor C. García +3

Let $\mathcal{H}$ be a multiplicative subgroup of ${\mathbb{F}}_p^{\ast }$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \…