Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Published electronically: October 31, 2000
MathSciNet review: 1814088
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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

References [Enhancements On Off] (What's this?)

  • 1. R. Canetti, J. B. Friedlander, S. Konyagin, M. Larsen, D. Lieman and I. E. Shparlinski, `On the statistical properties of Diffie-Hellman distributions', Israel J. Math. (to appear).
  • 2. R. Canetti, J. B. Friedlander and I. E. Shparlinski, `On certain exponential sums and the distribution of Diffie-Hellman triples', J. London Math. Soc., 59 (1999), 799-812. MR 2000g:11079
  • 3. F. R. K. Chung, `Several generalizations of Weil sums', J. Number Theory, 49 (1994), 95-106. MR 95h:11085
  • 4. E. Dobrowolski and K. S. Williams, `An upper bound for the sum $\sum_{n=a+1}^{a+H} f(n)$ for a certain class of functions $f$', Proc. Amer. Math. Soc., 114 (1993), 29-35. MR 92c:11086
  • 5. J. Friedlander and H. Iwaniec, `Estimates for character sums', Proc. Amer. Math. Soc., 119 (1993), 363-372. MR 93k:11074
  • 6. H. Iwaniec and A. Sárközy, `On a multiplicative hybrid problem', J. Number Theory, 26 (1987), 89-95. MR 88f:11022
  • 7. S. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge, 1999. CMP 2000:05
  • 8. N. M. Korobov, `On the distribution of digits in periodic fractions', Matem. Sbornik, 89 (1972), 654-670 (in Russian). MR 54:12619
  • 9. N. M. Korobov, Exponential sums and their applications, Kluwer Acad. Publ., Dordrecht, 1992. MR 93a:11068
  • 10. H. Niederreiter, `Quasi-Monte Carlo methods and pseudo-random numbers', Bull. Amer. Math. Soc., 84 (1978), 957-1041. MR 80d:65016
  • 11. A. Sárközy, `On the distribution of residues of products of integers', Acta Math. Hungar., 49 (1987), 397-401. MR 88e:11004
  • 12. I. E. Shparlinski, `On the distribution of primitive and irreducible polynomials modulo a prime', Diskretnaja Matem., 1 (1989), no.1, 117-124 (in Russian).
  • 13. I. M. Vinogradov, Elements of Number Theory, Dover Publ., NY, 1954. MR 15:933e

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11L07, 11T23, 11L26

Retrieve articles in all journals with MSC (2000): 11L07, 11T23, 11L26

Additional Information

John B. Friedlander
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia

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

American Mathematical Society