On the projections of Ahlfors regular sets in the plane

TL;DR

This paper establishes a discretized projection theorem for Ahlfors regular sets in the plane, showing that for measures with certain decay properties, there exists a projection direction with a large projected size for large subsets.

Contribution

It provides a new $ ext{delta}$-discretized projection theorem for Ahlfors regular sets, extending classical projection results to a discretized setting with measure decay conditions.

Findings

Existence of a projection direction with large projected size.

Quantitative bounds on projections of Ahlfors regular sets.

Application of measure decay conditions to projection estimates.

Abstract

This paper contains the following $\delta$-discretised projection theorem for Ahlfors regular sets in the plane. For all $C,\epsilon > 0$ and $s \in [0,1]$, there exists $\kappa > 0$ such that the following holds for all $\delta > 0$ small enough. Let $\nu$ be a Borel probability measure on $S^{1}$ satisfying $\nu(B(x,r)) \leq Cr^{\epsilon}$ for all $x \in S^{1}$ and $r > 0$. Let $K \subset B(1) \subset \mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $\theta \in \mathrm{spt\,} \nu$ such that $$|\pi_{\theta}(F)|_{\delta} \geq \delta^{\epsilon - s}$$ for all $F \subset K$ with $|F|_{\delta} \geq \delta^{\kappa - s}$. Here $\pi_{\theta}(z) = \theta \cdot z$ for $z \in \mathbb{R}^{2}$.