On the distribution of Kloosterman sums

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For a prime p, we consider Kloosterman sums Kₚ(a) = ∑ₓ∈ ₚ* (2 πi (x + ax⁻¹)/p), a ∈ 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 Kₚ(a) when a runs through 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 ∈, v ∈ for any two sufficiently large sets, ₚ*. We also improve a recent bound on the nonlinearity of a Boolean function associated with the sequence of signs of Kloosterman sums.