Dimension-free estimates for discrete maximal functions and lattice points in high-dimensional spheres and balls with small radii

We prove that the discrete Hardy-Littlewood maximal function associated with Euclidean spheres with small radii has dimension-free estimates on ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$ for $p\in [2,\mathrm{\infty })\mathrm{.}$ This implies an analogous result for the Euclidean balls, thus making progress on a question of E.M. Stein from the mid 1990s. Our work provides the first dimension-free estimates for full discrete maximal functions related to spheres and balls without relying on comparisons with their continuous counterparts. An important part of our argument is a uniform (dimension-free) count of lattice points in high-dimensional spheres and balls with small radii. We also established a dimension-free estimate for a multi-parameter maximal function of a combinatorial nature, which is a new phenomenon and may be useful for studying similar problems in the future.