A conjecture for arithmetic spherical maximal functions

TL;DR

This paper proposes a conjecture to characterize the boundedness of sparse discrete spherical maximal functions and presents supporting theoretical results, addressing a long-standing open problem in harmonic analysis.

Contribution

It introduces a new conjecture for sparse spherical maximal functions and provides a theorem supporting this conjecture, advancing understanding in the field.

Findings

Formulated a conjecture characterizing boundedness of sparse spherical maximal functions.

Proved a theorem supporting the conjecture.

Addresses a 24-year open problem in discrete harmonic analysis.

Abstract

For 24 years, it has been an open problem to obtain improved bounds, for the maximal function over a sparse sequence of discrete spherical averages, going beyond the range for the full discrete spherical maximal function. I formulate a conjecture to characterize the boundedness of such maximal functions and state a theorem in support of it.