On the distribution of Kloosterman sums

2006/08/24 by Shparlinski, I. E.
#11L05 #11L26 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0608595

Abstract

For a prime p, we consider Kloosterman sums K_p(a) = ∑_{x∈ \Fp^*} exp(2 πi (x + ax^-1)/p), a ∈ \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 \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 ∈ \cU, v ∈ \cV for any two sufficiently large sets \cU, \cV ⊆ \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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