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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Double exponential sums over thin sets


Authors: John B. Friedlander and Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 129 (2001), 1617-1621
MSC (2000): Primary 11L07, 11T23; Secondary 11L26
DOI: https://doi.org/10.1090/S0002-9939-00-05921-9
Published electronically: October 31, 2000
MathSciNet review: 1814088
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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 $t$ 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 $\vert\,{\mathcal Y}\vert \ge p^{3/4+ \delta}$.


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Additional Information

John B. Friedlander
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: frdlndr@math.toronto.edu

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia
Email: igor@ics.mq.edu.au

DOI: https://doi.org/10.1090/S0002-9939-00-05921-9
Received by editor(s): September 16, 1999
Published electronically: October 31, 2000
Additional Notes: The first author was supported in part by NSERC grant A5123 and by an NEC grant to the Institute for Advanced Study.
The second author was supported in part by ARC grant A69700294.
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2000 American Mathematical Society