Estimates of Trigonometric Sums over Subgroups and Some of Their Applications

In this paper, we obtain new upper bounds for trigonometric sums over subgroups $\Gamma \subset \mathbb Z_{p}^{*}$ whose size belongs to $[p^{28/95},p^{182/487}]$. Using an approach due to Malykhin, we refine estimates of such sums in $\mathbb Z_{p^{r}}^{*}$ and apply them to the divisibility problem for Fermat quotients.