“…Initiated by work of Bourgain [7] in ergodic theory, research in this direction has continued to evolve into a standalone subfield of harmonic analysis following the pivotal work of Magyar, Stein and Wainger [39], where they considered the discrete analogue of the spherical maximal function. Several authors have proved maximal and/or improving inequalities for discrete operators over lattice points on surfaces of arithmetic interest; see [1,2,10,13,21,24,25,29,31,35,36,41] for some such results. A distinctive feature of such work is the interplay between analysis and number theory, as the arithmetic properties of the underlying discrete set play a central role when the analogous continuous operator involves curvature.…”