  Let $G_1,..., G_n \in \Fp[X_1,...,X_m]$ be $n$ polynomials in $m$ variables
over the finite field $\Fp$ of $p$ elements. A result of {\'E}. Fouvry and N.
M. Katz shows that under some natural condition, for any fixed $\varepsilon$
and sufficiently large prime $p$ the vectors of fractional parts $$
(\{\frac{G_1(\vec{x})}{p}},...,\{\frac{G_n(\vec{x})}{p}}), \qquad \vec{x} \in
\Gamma, $$ are uniformly distributed in the unit cube $[0,1]^n$ for any cube
$\Gamma \in [0, p-1]^m$ with the side length $h \ge p^{1/2} (\log p)^{1 +
\varepsilon}$. Here we use this result to show the above vectors remain
uniformly distributed, when $\vec{x}$ runs through a rather general set. We
also obtain new results about the distribution of solutions to system of
polynomial congruences.
