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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.16671 |
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| _version_ | 1866917035381882880 |
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| author | Du, Jiahan |
| author_facet | Du, Jiahan |
| contents | This paper investigates a refinement of Marstrand's projection theorem; more specifically, let $Π_t, t\in[0,1]$ be a family of $m$ dimensional subspaces of the Euclidean space $\mathbb{R}^n$ and let $P_t:\mathbb{R}^4\mapsto Π_t$ be the orthogonal projections onto $Π_t$. We hope to determine the conditions on $Π_t$ under which, for any Borel $A\subset\mathbb{R}^n$, $\dim_H P_t(A)=\min(m,\dim_H A)$ holds for almost every $t$. We propose a conjectured condition on $Π_t$ and provide partial progress towards its resolution. We first establish a version of the polynomial Wolff axiom, and then apply polynomial partitioning to derive a version of the $L^p$ Kakeya inequality. Finally, we use a discretization procedure to obtain the desired bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restricted Marstrand's projection theorem for general families of linear subspaces Du, Jiahan Classical Analysis and ODEs This paper investigates a refinement of Marstrand's projection theorem; more specifically, let $Π_t, t\in[0,1]$ be a family of $m$ dimensional subspaces of the Euclidean space $\mathbb{R}^n$ and let $P_t:\mathbb{R}^4\mapsto Π_t$ be the orthogonal projections onto $Π_t$. We hope to determine the conditions on $Π_t$ under which, for any Borel $A\subset\mathbb{R}^n$, $\dim_H P_t(A)=\min(m,\dim_H A)$ holds for almost every $t$. We propose a conjectured condition on $Π_t$ and provide partial progress towards its resolution. We first establish a version of the polynomial Wolff axiom, and then apply polynomial partitioning to derive a version of the $L^p$ Kakeya inequality. Finally, we use a discretization procedure to obtain the desired bound. |
| title | Restricted Marstrand's projection theorem for general families of linear subspaces |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2510.16671 |