Emmanuel Kowalski
This is an expository account of the proof of the theorem of Bourgain, Glibichuk and Konyagin which provides non-trivial bounds for exponential sums over very small multiplicative subgroups of prime finite fields.
This paper contains 7 sections, 14 theorems, 146 equations.
Theorem 1.1
Let $\gamma>0$ be a real number. There exists a real number $\nu>0$, depending only on $\gamma$, such that for any prime number $p$ and any subgroup $H\subset \mathbf{F}_p^{\times}$ with $|H|\geqslant p^{\gamma}$, we have for any $a\in\mathbf{F}_p^{\times}$, where the implied constant depends only on $\gamma$.
