A note on discrete spherical averages over sparse sequences

TL;DR

This paper constructs a specific increasing sequence of radii for which the associated discrete spherical maximal operators are bounded on  for all p>1 in dimensions five and higher, advancing understanding of sparse spherical averages.

Contribution

It provides an example of a sparse sequence where discrete spherical maximal operators are bounded on  for all p>1 in dimensions 5 and above, highlighting new boundedness conditions.

Findings

Boundedness of maximal operators on  for p>1

Existence of sparse sequences with bounded spherical averages

Dimension requirement n5 for boundedness

Abstract

This note presents an example of an increasing sequence $(\lambda_l)_{l=1}^\infty$ such that the maximal operators associated to normalized discrete spherical convolution averages \[ \sup_{l\geq 1}\frac{1}{r(\lambda_l)}\left|\sum_{|x|^2=\lambda_l}f(y-x)\right|\] for functions $f:\mathbb{Z}^n\to\mathbb{C}^n$ are bounded on $\ell^p$ for all $p>1$ when the ambient dimension $n$ is at least five.