Universal Sets for Projections

2024/11/24 by Jacob B. Fiedler, Fiedler, Jacob B., D. M. Stull +1 · 1 citation
Computer Science · Engineering · #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #Manufacturing Process and Optimization

paper · pdf · doi:10.48550/arxiv.2411.16001

Abstract

We investigate variants of Marstrand's projection theorem that hold for sets of directions and classes of sets in ℝ². We say that a set of directions D \subseteqS¹ is universal for a class of sets if, for every set E in the class, there is a direction e∈ 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.

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