Length of sets under restricted families of projections onto lines
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909285339889664 |
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| author | Harris, Terence L. J. |
| author_facet | Harris, Terence L. J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_06896 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Length of sets under restricted families of projections onto lines Harris, Terence L. J. Classical Analysis and ODEs Analysis of PDEs 28A78, 28A80 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. |
| title | Length of sets under restricted families of projections onto lines |
| topic | Classical Analysis and ODEs Analysis of PDEs 28A78, 28A80 |
| url | https://arxiv.org/abs/2208.06896 |