On the distribution of values and zeros of polynomial systems over arbitrary sets

Let G1,...,Gn∈Fp[X1,...,Xm] be n polynomials in m variables over the finite field Fp of p elements. A result of É. Fouvry and N.M. Katz shows that under some natural condition, for any fixed ε and sufficiently large prime p the vectors of fractional parts({G1(x)p},...,{Gn(x)p}),x∈Γ, are uniformly distributed in the unit cube ^[0,1]n for any cube Γ ∈ ^{[0,p -1]m} with the side length h ≥ ^{p1/2(logp)1 +ε}. Here we use this result to show the above vectors remain uniformly distributed, when x runs through a rather general set. We also obtain new results about the distribution of solutions to system of polynomial congruences.