  For a prime $p$, we consider Kloosterman sums $$ K_{p}(a) = \sum_{x\in
\F_p^*} \exp(2 \pi i (x + ax^{-1})/p), \qquad a \in \F_p^*, $$ 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 $u \in \cU$, $v \in
\cV$ for any two sufficiently large sets $\cU, \cV \subseteq \F_p^*$.
  We also improve a recent bound on the nonlinearity of a Boolean function
associated with the sequence of signs of Kloosterman sums.
