DISTRIBUTION OF DISTANCES IN POSITIVE CHARACTERISTIC

Let F-q be an arbitrary finite field, and epsilon be a point set in F-q(d). Let Delta(epsilon) be the set of distances determined by pairs of points in E. Using Kloosterman sums, Iosevich and Rudnev (2007) proved that if vertical bar epsilon vertical bar >= 4q((d + 1)/2) then Delta(epsilon) = F-q. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. We use the point-plane incidence bound due to Rudnev to prove that if epsilon has Cartesian product structure in vector spaces over prime fields, then we can break the exponent (d + 1)/2 and still cover all distances. We also show that the number of pairs of points in epsilon of any given distance is close to its expected value.