Projections, Furstenberg sets, and the $ABC$ sum-product problem
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911532966739968 |
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| author | Orponen, Tuomas Shmerkin, Pablo |
| author_facet | Orponen, Tuomas Shmerkin, Pablo |
| contents | We make progress on two interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, and the dimension of Furstenberg sets. Along the way, we obtain new information on the dimension of exceptional sets of orthogonal projections.
First, we give a new proof of the following asymmetric sum-product theorem: Let $A,B,C \subset \mathbb{R}$ be Borel sets with $0 < {\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A < 1$ and ${\dim_{\mathrm{H}}} B + {\dim_{\mathrm{H}}} C > {\dim_{\mathrm{H}}} A$. Then, there exists $c \in C$ such that $$ \dim_{\mathrm{H}} (A + cB) > {\dim_{\mathrm{H}}} A. $$ We use this to show that every $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ associated with a line set of equal Hausdorff and packing dimension $t$ satisfies $$\dim_{\mathrm{H}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10199 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Projections, Furstenberg sets, and the $ABC$ sum-product problem Orponen, Tuomas Shmerkin, Pablo Classical Analysis and ODEs Combinatorics Metric Geometry 28A80, 28A78, 51A20 We make progress on two interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, and the dimension of Furstenberg sets. Along the way, we obtain new information on the dimension of exceptional sets of orthogonal projections. First, we give a new proof of the following asymmetric sum-product theorem: Let $A,B,C \subset \mathbb{R}$ be Borel sets with $0 < {\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A < 1$ and ${\dim_{\mathrm{H}}} B + {\dim_{\mathrm{H}}} C > {\dim_{\mathrm{H}}} A$. Then, there exists $c \in C$ such that $$ \dim_{\mathrm{H}} (A + cB) > {\dim_{\mathrm{H}}} A. $$ We use this to show that every $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ associated with a line set of equal Hausdorff and packing dimension $t$ satisfies $$\dim_{\mathrm{H}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}.$$ |
| title | Projections, Furstenberg sets, and the $ABC$ sum-product problem |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry 28A80, 28A78, 51A20 |
| url | https://arxiv.org/abs/2301.10199 |