Universal Sets for Projections
Jacob B. Fiedler, D. M. Stull

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 . We say that a set of directions is for a class of sets if, for every set in the class, there is a direction such that the projection of in the direction 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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Digital Image Processing Techniques · Manufacturing Process and Optimization
