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 | References | Similar Articles | Additional Information

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 $| {\mathcal Y}| \ge p^{3/4+ \delta }$.

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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

MR Author ID:
192194

Email:
igor@ics.mq.edu.au

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