Slices and distances: on two problems of Furstenberg and Falconer
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866929460509409280 |
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| author | Shmerkin, Pablo |
| author_facet | Shmerkin, Pablo |
| contents | We survey the history and recent developments around two decades-old problems that continue to attract a great deal of interest: the slicing $\times 2$, $\times 3$ conjecture of H. Furstenberg in ergodic theory, and the distance set problem in geometric measure theory introduced by K. Falconer. We discuss some of the ideas behind our solution of Furstenberg's slicing conjecture, and recent progress in Falconer's problem. While these two problems are on the surface rather different, we emphasize some common themes in our approach: analyzing fractals through a combinatorial description in terms of ``branching numbers'', and viewing the problems through a ``multiscale projection'' lens. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_12157 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Slices and distances: on two problems of Furstenberg and Falconer Shmerkin, Pablo Classical Analysis and ODEs Combinatorics Dynamical Systems Metric Geometry Primary: 11K55, 28A75, 28A80, 37C45, Secondary: 05D99, 28A78 We survey the history and recent developments around two decades-old problems that continue to attract a great deal of interest: the slicing $\times 2$, $\times 3$ conjecture of H. Furstenberg in ergodic theory, and the distance set problem in geometric measure theory introduced by K. Falconer. We discuss some of the ideas behind our solution of Furstenberg's slicing conjecture, and recent progress in Falconer's problem. While these two problems are on the surface rather different, we emphasize some common themes in our approach: analyzing fractals through a combinatorial description in terms of ``branching numbers'', and viewing the problems through a ``multiscale projection'' lens. |
| title | Slices and distances: on two problems of Furstenberg and Falconer |
| topic | Classical Analysis and ODEs Combinatorics Dynamical Systems Metric Geometry Primary: 11K55, 28A75, 28A80, 37C45, Secondary: 05D99, 28A78 |
| url | https://arxiv.org/abs/2109.12157 |