Dimension-free estimates for the discrete spherical maximal functions

TL;DR

This paper establishes dimension-free $ ext{L}^p$ bounds for discrete spherical maximal functions in high dimensions, using asymptotic formulas and new multiplier techniques to control growth in dimension.

Contribution

It introduces a novel approach combining asymptotic Waring problem formulas and multiplier approximations to achieve dimension-independent bounds for discrete spherical maximal functions.

Findings

Dimension-free $ ext{L}^p$ bounds for $p ext{ in } [2, ext{infinity}]$ in $ extbf{Z}^d$ for $d ext{ at least } 5$

Asymptotic formula with a multiplicative error term independent of dimension

New multiplier methods to control exponential growth in dimension

Abstract

We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein and Wainger) corresponding to the Euclidean spheres in $\mathbb Z^d$ with dyadic radii have $\ell^p(\mathbb Z^d)$ bounds for all $p\in[2, \infty]$ independent of the dimensions $d\ge 5$. 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.