Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912159319982080 |
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| author | Shmerkin, Pablo Wang, Hong |
| author_facet | Shmerkin, Pablo Wang, Hong |
| contents | We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+ε$, where $ε>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s<t\le 1+ε(s,t)$, in particular providing the first improvement since 1999 to the dimension of classical $s$-Furstenberg sets for $s<1/2$. We deduce this from a corresponding discretized incidence bound under minimal non-concentration assumptions, that simultaneously extends Bourgain's discretized projection and sum-product theorems. The proofs are based on a recent discretized incidence bound of T.~Orponen and the first author and a certain duality between $(s,t)$ and $(t/2,s+t/2)$-Furstenberg sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_13363 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem Shmerkin, Pablo Wang, Hong Classical Analysis and ODEs Combinatorics Metric Geometry 28A80 (Primary) 28A75, 28A78 (Secondary) We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+ε$, where $ε>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s<t\le 1+ε(s,t)$, in particular providing the first improvement since 1999 to the dimension of classical $s$-Furstenberg sets for $s<1/2$. We deduce this from a corresponding discretized incidence bound under minimal non-concentration assumptions, that simultaneously extends Bourgain's discretized projection and sum-product theorems. The proofs are based on a recent discretized incidence bound of T.~Orponen and the first author and a certain duality between $(s,t)$ and $(t/2,s+t/2)$-Furstenberg sets. |
| title | Dimensions of Furstenberg sets and an extension of Bourgain's projection theorem |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry 28A80 (Primary) 28A75, 28A78 (Secondary) |
| url | https://arxiv.org/abs/2211.13363 |