Restricted weak-type endpoint estimates for k-spherical maximal functions

In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the l(p)(Z(d))-boundedness results for discrete k-spherical maximal functions to a restricted weak-type result at the endpoint. We introduce a density-parameter, which may be viewed as a discrete version of Minkowski dimension used in related works on the continuous analgoue, in order to exploit recent progress of Wooley and Bourgain-Demeter-Guth on the Vinogradov mean value conjectures via a novel Approximation Formula for a single average, and obtain improved bounds for lacunary discrete k-spherical maximal functions when k >= 3.