Intersections of thick compact sets in $\mathbb{R}^d$
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2021
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912844008652800 |
|---|---|
| author | Falconer, Kenneth Yavicoli, Alexia |
| author_facet | Falconer, Kenneth Yavicoli, Alexia |
| contents | We introduce a definition of thickness in $\mathbb{R}^d$ and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in $\mathbb{R}^d$ with thickness $τ$, there is a number $N(τ)$ such that the set contains a translate of all sufficiently small similar copies of every set in $\mathbb{R}^d$ with at most $N(τ)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_01186 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Intersections of thick compact sets in $\mathbb{R}^d$ Falconer, Kenneth Yavicoli, Alexia Classical Analysis and ODEs Dynamical Systems 11B25, 28A12, 28A78, 28A80 We introduce a definition of thickness in $\mathbb{R}^d$ and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in $\mathbb{R}^d$ with thickness $τ$, there is a number $N(τ)$ such that the set contains a translate of all sufficiently small similar copies of every set in $\mathbb{R}^d$ with at most $N(τ)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness. |
| title | Intersections of thick compact sets in $\mathbb{R}^d$ |
| topic | Classical Analysis and ODEs Dynamical Systems 11B25, 28A12, 28A78, 28A80 |
| url | https://arxiv.org/abs/2102.01186 |