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arXiv · 2411.04959

Bounding the dimension of exceptional sets for orthogonal projections

Peter Cholak·Marianna Csornyei·Neil Lutz·Patrick Lutz·Elvira Mayordomo·D. M. Stull
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Abstract

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(π_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$ and $π_V$ is the orthogonal projection of $A$ onto $V$. In this paper we study how large the exceptional set \begin{equation*} \{V\in G(n,k) \mid \dim_H(π_V A) < s\} \end{equation*} can be for a given $s\le\min\{a,k\}.$ 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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Peter Cholak, Marianna Csornyei, Neil Lutz, Patrick Lutz, Elvira Mayordomo, D. M. Stull. 2025-09-02. Bounding the dimension of exceptional sets for orthogonal projections. https://arxiv.org/abs/2411.04959

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