The discrete spherical averages over a family of sparse sequences

TL;DR

This paper investigates the boundedness of discrete spherical maximal functions over sparse sequences in integer lattices, proving new bounds and establishing endpoint results using advanced number-theoretic techniques.

Contribution

It introduces new bounds for sparse sequences, including an endpoint result in four dimensions, and develops refined methods involving Kloosterman sums and spherical measure decomposition.

Findings

Proves boundedness of spherical maximal function in Z^4 over sparse radii

Establishes new bounds for sparse sequences achieving endpoint of Magyar--Stein--Wainger theorem

Utilizes Kloosterman refinement and Weil bounds for advanced analysis

Abstract

We initiate the study of the $\ell^p(\mathbb{Z}^d)$-boundedness of the arithmetic spherical maximal function over sparse sequences. We state a folklore conjecture for lacunary sequences, a key example of Zienkiewicz and prove new bounds for a family of sparse sequences that achieves the endpoint of the Magyar--Stein--Wainger theorem for the full discrete spherical maximal function. Perhaps our most interesting result is the boundedness of a discrete spherical maximal function in $\mathbb{Z}^4$ over an infinite, albeit sparse, set of radii. Our methods include the Kloosterman refinement for the Fourier transform of the spherical measure (introduced by Magyar) and Weil bounds for Kloosterman sums which are utilized by a new further decomposition of spherical measure.