Dimension-Free Estimates for the Discrete Spherical Maximal Functions

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