On the uniformity and size of microsets
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912661420113920 |
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| author | Balka, Richárd Orgoványi, Vilma Rutar, Alex |
| author_facet | Balka, Richárd Orgoványi, Vilma Rutar, Alex |
| contents | We resolve a few questions regarding the uniformity and size of microsets of subsets of Euclidean space. First, we construct a compact set $K\subset\mathbb{R}^d$ with Assouad dimension arbitrarily close to $d$ such that every microset of $K$ has no Ahlfors--David regular subset with dimension strictly larger than $0$. This answers a question of Orponen. Then, we show that for any non-empty compact set $K\subset\mathbb{R}^d$ with lower dimension $β$, there is a microset $E$ of $K$ with finite $β$-dimensional packing pre-measure. This answers a strong version of a question of Fraser--Howroyd--Käenmäki--Yu, who previously obtained a similar result concerning the upper box dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20594 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the uniformity and size of microsets Balka, Richárd Orgoványi, Vilma Rutar, Alex Metric Geometry Dynamical Systems 28A80 (Primary) 28A78 (Secondary) We resolve a few questions regarding the uniformity and size of microsets of subsets of Euclidean space. First, we construct a compact set $K\subset\mathbb{R}^d$ with Assouad dimension arbitrarily close to $d$ such that every microset of $K$ has no Ahlfors--David regular subset with dimension strictly larger than $0$. This answers a question of Orponen. Then, we show that for any non-empty compact set $K\subset\mathbb{R}^d$ with lower dimension $β$, there is a microset $E$ of $K$ with finite $β$-dimensional packing pre-measure. This answers a strong version of a question of Fraser--Howroyd--Käenmäki--Yu, who previously obtained a similar result concerning the upper box dimension. |
| title | On the uniformity and size of microsets |
| topic | Metric Geometry Dynamical Systems 28A80 (Primary) 28A78 (Secondary) |
| url | https://arxiv.org/abs/2412.20594 |