Sparse bounds for discrete singular Radon transforms

TL;DR

This paper establishes sparse bounds for discrete singular Radon transforms along polynomial mappings, extending results to all polynomials in one dimension and identifying restrictions in higher dimensions due to geometric, analytic, and number-theoretic challenges.

Contribution

It provides the first sparse bounds for a broad class of discrete singular Radon transforms, including all polynomials in one dimension and outlining conditions in higher dimensions.

Findings

Sparse bounds proven for all polynomials in one dimension.

Restrictions identified for polynomial mappings in higher dimensions.

Highlights interplay of geometry, analysis, and number theory in the problem.

Abstract

We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:\mathbb{Z}^d\to \mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.