On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866914910331469824 |
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| author | Shmerkin, Pablo Wang, Hong |
| author_facet | Shmerkin, Pablo Wang, Hong |
| contents | We establish the dimension version of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions $d=2$ or $3$, we obtain the first explicit estimates for the dimensions of distance sets of general Borel sets of dimension $d/2$; for example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension $1$ has Hausdorff dimension at least $(\sqrt{5}-1)/2\approx 0.618$. In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension $d/2$. These results rely on new estimates for the dimensions of radial projections that may have independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_09044 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$ Shmerkin, Pablo Wang, Hong Classical Analysis and ODEs Combinatorics Metric Geometry Primary: 28A78, 28A80 We establish the dimension version of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions $d=2$ or $3$, we obtain the first explicit estimates for the dimensions of distance sets of general Borel sets of dimension $d/2$; for example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension $1$ has Hausdorff dimension at least $(\sqrt{5}-1)/2\approx 0.618$. In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension $d/2$. These results rely on new estimates for the dimensions of radial projections that may have independent interest. |
| title | On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$ |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry Primary: 28A78, 28A80 |
| url | https://arxiv.org/abs/2112.09044 |