Universal Sets for Projections

TL;DR

This paper explores the existence of small universal sets of directions in the plane that guarantee maximal Hausdorff dimension projections for various classes of sets, extending Marstrand's projection theorem.

Contribution

It constructs small universal sets for different classes of sets, including weakly regular and AD-regular sets, highlighting the role of regularity in projection properties.

Findings

Existence of universal sets with arbitrarily small positive Hausdorff dimension for weakly regular sets.

Existence of a universal set with zero Hausdorff dimension for AD-regular sets.

Extension of Marstrand's projection theorem to specific classes of sets and directions.

Abstract

We investigate variants of Marstrand's projection theorem that hold for sets of directions and classes of sets in $\mathbb{R}^2$. We say that a set of directions $D \subseteq\mathcal{S}^1$ is $\textit{universal}$ for a class of sets if, for every set $E$ in the class, there is a direction $e\in D$ such that the projection of $E$ in the direction $e$ has maximal Hausdorff dimension. We construct small universal sets for certain classes. Particular attention is paid to the role of regularity. We prove the existence of universal sets with arbitrarily small positive Hausdorff dimension for the class of weakly regular sets. We prove that there is a universal set of zero Hausdorff dimension for the class of AD-regular sets.