The Distribution of Solutions to Equations over Finite Fields

Let F be the finite field in q = p1 elements, £(x) be a A:-tuple of polynomials in F [xx,..., x„], V be the set of points in ¥'j satisfying F(x) = 0 and S,T be any subsets of F;. Set y)= L e{— TrU '?)) íory±Q, and 4>(K) = maxv\ 02( V)q2k. In case q = p v/e deduce from this, for example, that if C is a convex subset of R" symmetric about a point in Z", of diameter < 2p (with respect to the sup norm), and Vol(C) > 22"<&(V)pk, then C contains a solution of F(x) a 0(mod/>). We also show that if S is a box of points in ¥¡j not contained in any (n l)-dimensional subspace and |Z3| > 4 • 2"f<b(V)qk, then B r\ V contains n linearly independent points.