Restricted Radon transforms and projections of planar sets

TL;DR

This paper proves a new estimate for the Radon transform in the plane with fractional dimension directions, and applies it to study exceptional projection directions of planar sets, leading to a conjecture similar to Furstenberg's.

Contribution

It introduces a mixed norm estimate for the Radon transform with fractional dimension directions and explores its implications for projections of planar sets, proposing a related conjecture.

Findings

Established a mixed norm estimate for the Radon transform with fractional directions.

Identified an exceptional set of directions related to planar set projections.

Formulated a conjecture analogous to Furstenberg's based on these results.

Abstract

We establish a mixed norm estimate for the Radon transform in the plane when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.