On discrete fractional integral operators and related Diophantine equations

We study discrete versions of fractional integral operators along curves and surfaces. l(p) -> l(q) estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the discrete fractional integral along the curve gamma(m) = (m, m(2), . . . , m(k)) to Vinogradov's mean value theorem. Sharp l(p) -> l(q) estimates of the discrete fractional integral along the hyperbolic paraboloid in Z(3) are also obtained except for endpoints.