Dimension-Free Estimates for the Discrete Spherical Maximal Functions

We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein, and Wainger) corresponding to the Euclidean spheres in Z(d) with dyadic radii have l(p)(Z(d)) bounds for all p is an element of [2, infinity] independent of the dimensions d >= 5. An important part of our argument is the asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term. By considering new approximating multipliers, we will show how to absorb an exponential in dimension (like C-d for some C > 1) growth in norms arising from the sampling principle of Magyar, Stein, and Wainger and ultimately deduce dimension-free estimates for the discrete spherical maximal functions.

Automated summary

We prove that the discrete spherical maximal functions corresponding to the Euclidean spheres in Z(d) with dyadic radii have bounded norms in l(p)(Z(d)) for all p in [2, infinity], independent of the dimensions d >= 5. The asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term plays a crucial role in our argument. By introducing new approximating multipliers, we demonstrate how to absorb exponential growth in norms arising from the sampling principle and obtain dimension-free estimates for the discrete spherical maximal functions.