## On the distribution of Kloosterman sums

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- by Igor E. Shparlinski PDF
- Proc. Amer. Math. Soc.
**136**(2008), 419-425 Request permission

## Abstract:

For a prime $p$, we consider Kloosterman sums \[ K_{p}(a) = \sum _{x\in \mathbb {F}_p^*} \exp (2 \pi i (x + ax^{-1})/p), \qquad a \in \mathbb {F}_p^*,\] 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.## References

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## Additional Information

**Igor E. Shparlinski**- Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
- MR Author ID: 192194
- Email: igor@ics.mq.edu.au
- 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
- © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2358479