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In this paper, we use the Approximation Formula for the Fourier transform ofthe solution set of lattice points on k-spheres and methods of Bourgain andIonescu to refine the l^p(Z^d)-boundedness results for discrete k- sphericalmaximal functions to a restricted weak-type result at the endpoint. Moreover weintroduce a novel Approximation Formula for a single average; this allows us toimprove our bounds for discrete k-spherical maximal functions along sparsesubsequences of radii by exploiting recent progress of Wooley on the Vinogradovmean value conjectures. In particular we have improved bounds for lacunarydiscrete k-spherical maximal functions when k>2. We introduce adensity-parameter, which may be viewed as a discrete version of Minkowskidimension used in related works on the conitnuous phenomena, to prove ourresults for sparse averages.

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