Length of sets under restricted families of projections onto lines

Let \(\gamma: I \to S^2\) be a \(C^2\) curve with \(\det(\gamma, \gamma', \gamma'')\) nonvanishing, and for each \(\theta \in I\) let \(\rho_{\theta}\) be orthogonal projection onto the span of \(\gamma(\theta)\). It is shown that if \(A \subseteq \mathbb{R}^3\) is a Borel set of Haus...

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Bibliographic Details
Published in:arXiv.org 2023-03
Main Author: Harris, Terence L J
Format: Article
Language:English
Subjects:
Borel sets
Online Access:Get full text
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Summary:Let \(\gamma: I \to S^2\) be a \(C^2\) curve with \(\det(\gamma, \gamma', \gamma'')\) nonvanishing, and for each \(\theta \in I\) let \(\rho_{\theta}\) be orthogonal projection onto the span of \(\gamma(\theta)\). It is shown that if \(A \subseteq \mathbb{R}^3\) is a Borel set of Hausdorff dimension strictly greater than 1, then \(\rho_{\theta}(A)\) has positive length for a.e. \(\theta \in I\). This answers a question raised by K\"aenm\"aki, Orponen and Venieri.
ISSN:2331-8422
DOI:10.48550/arxiv.2208.06896