Double exponential sums over thin sets

Abstract:

We estimate double exponential sums of the form \begin{equation*} S_a( {\mathcal X}, {\mathcal Y}) = \sum _{x \in {\mathcal X}} \sum _{y \in {\mathcal Y}} \exp \left ( 2\pi i a \vartheta ^{xy}/p\right ), \end{equation*} where $\vartheta$ is of multiplicative order $t$ modulo the prime $p$ and ${\mathcal X}$ and ${\mathcal Y}$ are arbitrary subsets of the residue ring modulo $t$. In the special case $t = p-1$, our bound is nontrivial for $| {\mathcal X}| \ge | {\mathcal Y}| \ge p^{15/16+ \delta }$ with any fixed $\delta >0$, while if in addition we have $| {\mathcal X}| \ge p^{1- \delta /4}$ it is nontrivial for $| {\mathcal Y}| \ge p^{3/4+ \delta }$.