Projections of planar sets in well-separated directions

TL;DR

This paper investigates the complexity of projections of planar sets in well-separated directions, providing sharp bounds using the polynomial method and analyzing projections of Ahlfors-David regular sets.

Contribution

It introduces new bounds on the covering numbers of projections in well-separated directions and constructs examples showing sharpness, advancing understanding of geometric projections.

Findings

New bounds on the number of well-separated directions with small projections

Construction of examples demonstrating sharpness of bounds

Proven lower bounds on the dimension of projections of Ahlfors-David regular sets

Abstract

First, let $K \subset B(0,1) \subset \mathbb{R}^{2}$ be a set with $\mathcal{H}_{\infty}^{1}(K) \sim 1$, and write $\pi_{e}(K)$ for the orthogonal projection of $K$ into the line spanned by $e \in S^{1}$. For $1/2 \leq s < 1$, write $$E_{s} := \{e : N(\pi_{e}(K),\delta) \leq \delta^{-s}\}, $$ where $N(A,r)$ is the $r$-covering number of the set $A$. It is well-known -- and essentially due to R. Kaufman -- that $N(E_{s},\delta) \lessapprox \delta^{-s}$. Using the polynomial method, I prove that $$ N(E_{s},r) \lessapprox \min\left\{\delta^{-s}\left(\frac{\delta}{r}\right)^{1/2},r^{-1}\right\}, \quad \delta \leq r \leq 1.$$ I construct examples showing that the exponents in the bound are sharp for $\delta \leq r \leq \delta^{s}$. The second theorem concerns projections of $1$-Ahlfors-David regular sets. Let $A \geq 1$ and $1/2 \leq s < 1$ be given. I prove that, for $p = p(A,s) \in \mathbb{N}$ large enough, the finite set of unit vectors $S_{p} := \{e^{2\pi i k/p} : 0 \leq k < p\}$ has the following property. If $K \subset B(0,1)$ is non-empty and $1$-Ahlfors-David regular with regularity constant at most $A$, then $$\frac{1}{p} \sum_{e \in S_{p}} N(\pi_{e}(K),\delta) \geq \delta^{-s}$$ for all small enough $\delta > 0$. In particular, $\overline{\dim}_{\text{B}} \pi_{e}(K) \geq s$ for some $e \in S_{p}$.