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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 $γ: I \to S^2$ be a $C^2$ curve with $\det(γ, γ', γ'')$ nonvanishing, and for each $θ\in I$ let $ρ_θ$ be orthogonal projection onto the span of $γ(θ)$. It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension strictly greater than 1, then $ρ_θ(A)$ has positive length for a.e. $θ\in I$. This answers a question raised by Käenmäki, Orponen and Venieri.

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Terence L. J. Harris. 2023-03-14. Length of sets under restricted families of projections onto lines. https://doi.org/10.1090/conm%2F792%2F15892

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