On the projections of Ahlfors regular sets in the plane
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929540181262336 |
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| author | Orponen, Tuomas |
| author_facet | Orponen, Tuomas |
| contents | This paper contains the following $δ$-discretised projection theorem for Ahlfors regular sets in the plane.
For all $C,ε> 0$ and $s \in [0,1]$, there exists $κ> 0$ such that the following holds for all $δ> 0$ small enough. Let $ν$ be a Borel probability measure on $S^{1}$ satisfying $ν(B(x,r)) \leq Cr^ε$ for all $x \in S^{1}$ and $r > 0$. Let $K \subset B(1) \subset \mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $θ\in \mathrm{spt\,} ν$ such that $$|π_θ(F)|_δ \geq δ^{ε- s}$$ for all $F \subset K$ with $|F|_δ \geq δ^{κ- s}$. Here $π_θ(z) = θ\cdot z$ for $z \in \mathbb{R}^{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06872 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the projections of Ahlfors regular sets in the plane Orponen, Tuomas Classical Analysis and ODEs 27A80, 28A78 This paper contains the following $δ$-discretised projection theorem for Ahlfors regular sets in the plane. For all $C,ε> 0$ and $s \in [0,1]$, there exists $κ> 0$ such that the following holds for all $δ> 0$ small enough. Let $ν$ be a Borel probability measure on $S^{1}$ satisfying $ν(B(x,r)) \leq Cr^ε$ for all $x \in S^{1}$ and $r > 0$. Let $K \subset B(1) \subset \mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $θ\in \mathrm{spt\,} ν$ such that $$|π_θ(F)|_δ \geq δ^{ε- s}$$ for all $F \subset K$ with $|F|_δ \geq δ^{κ- s}$. Here $π_θ(z) = θ\cdot z$ for $z \in \mathbb{R}^{2}$. |
| title | On the projections of Ahlfors regular sets in the plane |
| topic | Classical Analysis and ODEs 27A80, 28A78 |
| url | https://arxiv.org/abs/2410.06872 |