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Double exponential sums over thin sets

Authors:	John B Friedlander Igor E Shparlinski

Institution:	Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3 ; Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia

Abstract:	We estimate double exponential sums of the form $\begin{equation}S_a(\,{\mathcal X},\,{\mathcal Y}) = \sum_{x \in \,{\mathcal X}... ...\in \,{\mathcal Y}} \exp\left( 2\pi i a \vartheta^{xy}/p\right), \end{equation}$ where $\vartheta$ is of multiplicative order modulo the prime and $\,{\mathcal X}$ and $\,{\mathcal Y}$ are arbitrary subsets of the residue ring modulo . In the special case , 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 $\vert\,{\mathcal Y}\vert \ge p^{3/4+ \delta}$ .

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