New estimates of double trigonometric sums with exponential functions

Abstract

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}\gamma(y)\exp(2\pi i a \lambda^{xy}/p)\Big|, \end{eqnarray*} where $\lambda$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $\gamma(n)$ are any complex numbers with $|\gamma(n)| \le 1.$ In particular, we improve several previously known bounds.