Double exponential sums over thin sets

We estimate double exponential sums of the form

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 $\vert\,{\mathcal X}\vert \ge \vert\,{\mathcal Y}\vert \ge p^{15/16+ \delta}$ with any fixed $\delta >0$, while if in addition we have $\vert\,{\mathcal X}\vert \ge p^{1- \delta/4}$ it is nontrivial for $\vert\,{\mathcal Y}\vert \ge p^{3/4+ \delta}$.