A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in $\mathbb{R}^3$

TL;DR

This paper proves a conjecture on the dimension of exceptional projections in three-dimensional space by establishing sharp bounds for a maximal function over fractal sets, using novel geometric techniques.

Contribution

It introduces new sharp $L^p$ bounds for a Wolff circular maximal function variant and applies lens cutting techniques from discrete geometry to solve a conjecture in geometric measure theory.

Findings

Resolved a conjecture on projection dimensions in $\

Established sharp $L^p$ bounds for a maximal function over fractals.

Developed lens cutting techniques from discrete geometry for harmonic analysis applications.

Abstract

We resolve a conjecture of F\"assler and Orponen on the dimension of exceptional projections to one-dimensional subspaces indexed by a space curve in $\mathbb{R}^3$. We do this by obtaining sharp $L^p$ bounds for a variant of the Wolff circular maximal function over fractal sets for a class of $C^2$ curves related to Sogge's cinematic curvature condition. A key new tool is the use of lens cutting techniques from discrete geometry.