Restriction estimates of epsilon-removal type for k-th powers and paraboloids

摘要

We obtain restriction estimates of e-removal type for the set of k-th powers of integers, and for discrete d-dimensional surfaces of the form which we term 'k-paraboloids'. For these surfaces, we obtain a satisfying range of exponents for large values of d, k. We also obtain estimates of e-removal type in the full supercritical range for k-th powers and for k-paraboloids of dimension d k(k-2). We rely on a variety of techniques in discrete harmonic analysis originating in Bourgain's works on the restriction theory of the squares and the discrete parabola.