Universal Sets for Projections
Jacob B. Fiedler, D. M. Stull
TL;DR
This work investigates universal sets of directions for projections in the plane, asking when a tiny subset of directions suffices to achieve maximal projection dimension for entire classes of sets. Using an effective-dimension framework, it proves extremely small universal sets exist for regular classes: a zero-dimensional universal set for AD-regular sets and arbitrarily small-dimension universal sets for weakly regular sets, with refinements tying to effective Hausdorff and packing dimensions. It also develops Bourgain-type universal sets, including a zero-dimensional Bourgain universal set for analytic objects and epsilon-dimension universals for sets with optimal oracles, connected to sharp exceptional-set bounds. The results illuminate how regularity assumptions enable strong universality phenomena and demonstrate methods to construct and control universal sets via combinatorial and algorithmic (Kolmogorov-complexity) techniques. Overall, the paper advances the restricted projection landscape by providing explicit, small universal sets across multiple regularity regimes and by linking universal behavior to algorithmic complexity through the point-to-set principle.
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.