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

We obtain restriction estimates of ε-removal type for the set of k-th powers of integers, and for discrete d-dimensional surfaces of the form {(n1,⋯,nd,n1k+⋯+ndk):|n1|,⋯,|nd|⩽N},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 ε-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.