Bounds on the dimension of lineal extensions
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913724721266688 |
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| author | Bushling, Ryan E. G. Fiedler, Jacob B. |
| author_facet | Bushling, Ryan E. G. Fiedler, Jacob B. |
| contents | Let $E \subseteq \mathbb{R}^n$ be a union of line segments and $F \subseteq \mathbb{R}^n$ the set obtained from $E$ by extending each line segment in $E$ to a full line. Keleti's line segment extension conjecture posits that the Hausdorff dimension of $F$ should equal that of $E$. Working in $\mathbb{R}^2$, we use effective methods to prove a strong packing dimension variant of this conjecture, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16315 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on the dimension of lineal extensions Bushling, Ryan E. G. Fiedler, Jacob B. Classical Analysis and ODEs 03D32, 28A80 (Primary) 68Q30 (Secondary) Let $E \subseteq \mathbb{R}^n$ be a union of line segments and $F \subseteq \mathbb{R}^n$ the set obtained from $E$ by extending each line segment in $E$ to a full line. Keleti's line segment extension conjecture posits that the Hausdorff dimension of $F$ should equal that of $E$. Working in $\mathbb{R}^2$, we use effective methods to prove a strong packing dimension variant of this conjecture, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems. |
| title | Bounds on the dimension of lineal extensions |
| topic | Classical Analysis and ODEs 03D32, 28A80 (Primary) 68Q30 (Secondary) |
| url | https://arxiv.org/abs/2404.16315 |