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arXiv · 2208.06896

Length of sets under restricted families of projections onto lines

Terence L. J. Harris
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Abstract

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.

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Terence L. J. Harris. 2022-08-14. Length of sets under restricted families of projections onto lines. https://doi.org/10.1090/conm%2F792%2F15892

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