Bounding the dimension of exceptional sets for orthogonal projections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909764394418176 |
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| author | Cholak, Peter Csornyei, Marianna Lutz, Neil Lutz, Patrick Mayordomo, Elvira Stull, D. M. |
| author_facet | Cholak, Peter Csornyei, Marianna Lutz, Neil Lutz, Patrick Mayordomo, Elvira Stull, D. M. |
| contents | 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 $n=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04959 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounding the dimension of exceptional sets for orthogonal projections Cholak, Peter Csornyei, Marianna Lutz, Neil Lutz, Patrick Mayordomo, Elvira Stull, D. M. Classical Analysis and ODEs 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 $n=3$. |
| title | Bounding the dimension of exceptional sets for orthogonal projections |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2411.04959 |