On restricted projections to planes in $\mathbb{R}^3$
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910380414992384 |
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| author | Gan, Shengwen Guo, Shaoming Guth, Larry Harris, Terence L. J. Maldague, Dominique Wang, Hong |
| author_facet | Gan, Shengwen Guo, Shaoming Guth, Larry Harris, Terence L. J. Maldague, Dominique Wang, Hong |
| contents | Let $γ:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(γ(θ),γ'(θ),γ"(θ)\big)\neq 0$. For each $θ\in[0,1]$, let $V_θ=γ(θ)^\perp$ and let $π_θ:\mathbb{R}^3\rightarrow V_θ$ be the orthogonal projections. We prove that if $A\subset \mathbb{R}^3$ is a Borel set, then for a.e. $θ\in [0,1]$ we have $\text{dim}(π_θ(A))=\min\{2,\text{dim} A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(π_θ(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+s-\text{dim}(A)$. We also prove that if $\text{dim}(A)>2$, then for a.e. $θ\in[0,1]$ we have $\mathcal{H}^2(π_θ(A))>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13844 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On restricted projections to planes in $\mathbb{R}^3$ Gan, Shengwen Guo, Shaoming Guth, Larry Harris, Terence L. J. Maldague, Dominique Wang, Hong Classical Analysis and ODEs Metric Geometry Let $γ:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(γ(θ),γ'(θ),γ"(θ)\big)\neq 0$. For each $θ\in[0,1]$, let $V_θ=γ(θ)^\perp$ and let $π_θ:\mathbb{R}^3\rightarrow V_θ$ be the orthogonal projections. We prove that if $A\subset \mathbb{R}^3$ is a Borel set, then for a.e. $θ\in [0,1]$ we have $\text{dim}(π_θ(A))=\min\{2,\text{dim} A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(π_θ(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+s-\text{dim}(A)$. We also prove that if $\text{dim}(A)>2$, then for a.e. $θ\in[0,1]$ we have $\mathcal{H}^2(π_θ(A))>0$. |
| title | On restricted projections to planes in $\mathbb{R}^3$ |
| topic | Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2207.13844 |