Packing dimension of vertical projections in the Heisenberg group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910044229992448 |
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| author | Harris, Terence L. J. |
| author_facet | Harris, Terence L. J. |
| contents | It is shown that if $A$ is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying $2< \dim A < 3$, then the packing dimensions of vertical projections of $A$ are almost surely not less than $\dim A$, where both packing and Hausdorff dimensions are defined with respect to the Korányi metric.
For the Hausdorff dimension of the projections, a weaker almost sure lower bound is obtained which improves the known bound in the range $2 < \dim A < \frac{1}{8}\left( 17 + \sqrt{33}\right) \approx 2.84$. The bound is slightly larger than $1+\frac{1}{2} \dim A$ and behaves similarly near $\dim A =2$. Both proofs rely on a variable coefficient local smoothing inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_11475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Packing dimension of vertical projections in the Heisenberg group Harris, Terence L. J. Classical Analysis and ODEs Metric Geometry 28A78, 28A80 It is shown that if $A$ is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying $2< \dim A < 3$, then the packing dimensions of vertical projections of $A$ are almost surely not less than $\dim A$, where both packing and Hausdorff dimensions are defined with respect to the Korányi metric. For the Hausdorff dimension of the projections, a weaker almost sure lower bound is obtained which improves the known bound in the range $2 < \dim A < \frac{1}{8}\left( 17 + \sqrt{33}\right) \approx 2.84$. The bound is slightly larger than $1+\frac{1}{2} \dim A$ and behaves similarly near $\dim A =2$. Both proofs rely on a variable coefficient local smoothing inequality. |
| title | Packing dimension of vertical projections in the Heisenberg group |
| topic | Classical Analysis and ODEs Metric Geometry 28A78, 28A80 |
| url | https://arxiv.org/abs/2407.11475 |