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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
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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Additional Information

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

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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