Bounding the dimension of exceptional sets for orthogonal projections

2024/11/07 by Cholak, Peter, Csornyei, Marianna, Lutz, Neil +3 · 2 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.04959

Abstract

It is well known that if A ⊆ ℝⁿ is an analytic set of Hausdorff dimension a, then dim_H(π_VA)=min\a,k\ for a.e. V∈ G(n,k), where G(n,k) denotes the set of all k-dimensional subspaces of ℝⁿ and π_V is the orthogonal projection of A onto V. In this paper we study how large the exceptional set \V∈ G(n,k) | dim_H(π_V A) lt; s\ can be for a given s≤min\a,k\. We improve previously known estimates on the dimension of the exceptional set, and we show that our estimates are sharp for k=1 and for k=n-1. Hence we completely resolve this question for n=3.

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