The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914085207015424 |
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| author | Craig, Sam |
| author_facet | Craig, Sam |
| contents | A $(1,2t)$-Furstenberg set in $\mathbb{R}^3$ is naturally defined as a set containing a union of unit line segments forming a $2t$-dimensional subset of the affine Grassmannian in $\mathbb{R}^3$ and satisfying a suitable variant of the Frostman Convex Wolff Axiom. Some of these sets have a multi-scale self-similarity property called stickiness. We investigate the extremizers of the quasi-Assouad dimension of $(1,2t)$-Furstenberg sets, a slightly stronger variant of the Assouad dimension. We prove that sticky $(1,2t)$-Furstenberg sets have the least possible quasi-Assouad dimension among all $(1,2t)$-Furstenberg sets.
This result also follows from Corollary 1.10 of Wang and Zahl's solution to the Kakeya conjecture, which implies that all $(1,2t)$-Furstenberg sets have Hausdorff dimension $2t+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06462 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets Craig, Sam Classical Analysis and ODEs A $(1,2t)$-Furstenberg set in $\mathbb{R}^3$ is naturally defined as a set containing a union of unit line segments forming a $2t$-dimensional subset of the affine Grassmannian in $\mathbb{R}^3$ and satisfying a suitable variant of the Frostman Convex Wolff Axiom. Some of these sets have a multi-scale self-similarity property called stickiness. We investigate the extremizers of the quasi-Assouad dimension of $(1,2t)$-Furstenberg sets, a slightly stronger variant of the Assouad dimension. We prove that sticky $(1,2t)$-Furstenberg sets have the least possible quasi-Assouad dimension among all $(1,2t)$-Furstenberg sets. This result also follows from Corollary 1.10 of Wang and Zahl's solution to the Kakeya conjecture, which implies that all $(1,2t)$-Furstenberg sets have Hausdorff dimension $2t+1$. |
| title | The quasi-Assouad dimension of $(1,2t)$-Furstenberg sets in $\mathbb{R}^3$ is extremized by sticky sets |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2510.06462 |