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On the distribution of Kloosterman sums


Author: Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 136 (2008), 419-425
MSC (2000): Primary 11L05, 11L40, 11T71
DOI: https://doi.org/10.1090/S0002-9939-07-08943-5
Published electronically: November 2, 2007
MathSciNet review: 2358479
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Abstract: For a prime $ p$, we consider Kloosterman sums

$\displaystyle K_{p}(a) = \sum_{x\in \mathbb{F}_p^*} \exp(2 \pi i (x + ax^{-1})/p), \qquad a \in \mathbb{F}_p^*,$

over a finite field of $ p$ elements. It is well known that due to results of Deligne, Katz and Sarnak, the distribution of the sums $ K_{p}(a)$ when $ a$ runs through $ \mathbb{F}_p^*$ is in accordance with the Sato-Tate conjecture. Here we show that the same holds where $ a$ runs through the sums $ a = u+v$ for $ u \in \mathcal{U}$, $ v \in \mathcal{V}$ for any two sufficiently large sets $ \mathcal{U}, \mathcal{V}\subseteq \mathbb{F}_p^*$.

We also improve a recent bound on the nonlinearity of a Boolean function associated with the sequence of signs of Kloosterman sums.


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  • 1. A. Adolphson, `On the distribution of angles of Kloosterman sums', J. Reine Angew. Math. 395 (1989), 214-220. MR 983069 (90k:11109)
  • 2. W. Banks and I. E. Shparlinski, `Non-residues and primitive roots in Beatty sequences', Bull. Aust. Math. Soc. 73 (2006), 433-443. MR 2230651 (2007a:11118)
  • 3. W. Banks and I. E. Shparlinski, `Short character sums with Beatty sequences', Math. Res. Lett. 13 (2006), 539-547. MR 2250489
  • 4. W. Banks and I. E. Shparlinski, `Prime divisors in Beatty sequences', J. Number Theory 123 (2007), 413-425.
  • 5. C. Carlet and C. Ding, `Highly nonlinear mappings', J. Compl., 20 (2004), 205-244. MR 2067428 (2006d:94043)
  • 6. C.-L. Chai and W.-C. W. Li, `Character sums, automorphic forms, equidistribution, and Ramanujan graphs. I: The Kloosterman sum conjecture over function fields', Forum Math. 15 (2003), 679-699. MR 2010030 (2005h:11136)
  • 7. É. Fouvry and P. Michel, `Sommes de modules de sommes d'exponentielles', Pacific J. Math. 209 (2003), 261-288. MR 1978371 (2004d:11072)
  • 8. É. Fouvry and P. Michel, `Sur le changement de signe des sommes de Kloosterman', Ann. Math. (to appear).
  • 9. É. Fouvry, P. Michel, J. Rivat and A. Sárközy, `On the pseudorandomness of the signs of Kloosterman sums', J. Aust. Math. Soc. 77 (2004), 425-436. MR 2099811 (2005h:11165)
  • 10. H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
  • 11. N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Princeton Univ. Press, Princeton, NJ, 1988. MR 955052 (91a:11028)
  • 12. N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, Amer. Math. Soc, Providence, RI, 1999. MR 1659828 (2000b:11070)
  • 13. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience, New York-London-Sydney, 1974. MR 0419394 (54:7415)
  • 14. G. Laumon, `Exponential sums and $ l$-adic cohomology: A survey', Israel J. Math. 120 (2000), 225-257. MR 1815377 (2002m:11075)
  • 15. R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, Cambridge, 1997. MR 1429394 (97i:11115)
  • 16. P. Michel, `Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman, II', Duke Math. J. 92 (1998), 221-254. MR 1612781 (99d:11094)
  • 17. P. Michel, `Minoration de sommes d'exponentielles', Duke Math. J. 95 (1998), 227-240. MR 1652005 (99i:11069)
  • 18. H. Niederreiter, `The distribution of values of Kloosterman sums', Arch. Math. 56 (1991), 270-277. MR 1091880 (92b:11057)
  • 19. I. E. Shparlinski, `On the nonlinearity of the sequence of signs of Kloosterman sums', Bull. Aust. Math. Soc. 71 (2005), 405-409. MR 2150929 (2006e:11115)
  • 20. J. Vaaler, `Some extremal functions in Fourier analysis', Bull. Amer. Math. Soc. 12 (1985), 183-216. MR 776471 (86g:42005)

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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-07-08943-5
Received by editor(s): August 20, 2006
Received by editor(s) in revised form: September 29, 2006
Published electronically: November 2, 2007
Additional Notes: During the preparation of this paper, the author was supported in part by ARC grant DP0556431.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society

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