A nonlinear version of Bourgain's projection theorem
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866913468761767936 |
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| author | Shmerkin, Pablo |
| author_facet | Shmerkin, Pablo |
| contents | We prove a version of Bourgain's projection theorem for parametrized families of $C^2$ maps, that refines the original statement even in the linear case. As one application, we show that if $A$ is a Borel set of Hausdorff dimension close to $1$ in $\mathbb{R}^2$ or close to $3/2$ in $\mathbb{R}^3$, then for $y\in A$ outside of a very sparse set, the pinned distance set $\{|x-y|:x\in A\}$ has Hausdorff dimension at least $1/2+c$, where $c$ is universal. Furthermore, the same holds if the distances are taken with respect to a $C^2$ norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between $δ$-balls and $δ$-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into ``Frostman pieces'' that may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_01636 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A nonlinear version of Bourgain's projection theorem Shmerkin, Pablo Classical Analysis and ODEs Combinatorics Metric Geometry Primary: 28A75, 28A80, Secondary: 05D99, 26A16, 49Q15 We prove a version of Bourgain's projection theorem for parametrized families of $C^2$ maps, that refines the original statement even in the linear case. As one application, we show that if $A$ is a Borel set of Hausdorff dimension close to $1$ in $\mathbb{R}^2$ or close to $3/2$ in $\mathbb{R}^3$, then for $y\in A$ outside of a very sparse set, the pinned distance set $\{|x-y|:x\in A\}$ has Hausdorff dimension at least $1/2+c$, where $c$ is universal. Furthermore, the same holds if the distances are taken with respect to a $C^2$ norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between $δ$-balls and $δ$-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into ``Frostman pieces'' that may be of independent interest. |
| title | A nonlinear version of Bourgain's projection theorem |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry Primary: 28A75, 28A80, Secondary: 05D99, 26A16, 49Q15 |
| url | https://arxiv.org/abs/2003.01636 |