On the distribution of Kloosterman sums

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.