New results on embeddings of self-similar sets via renormalization
| Autor: | Algom, Amir, Hochman, Michael, Wu, Meng |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | For self-similar sets $X,Y\subseteq \mathbb{R}$, we obtain new results towards the affine embeddings conjecture of Feng-Huang-Rao (2014), and the equivalent weak intersections conjecture. We show that the conjecture holds when the defining maps of $X,Y$ have algebraic contraction ratios, and also for arbitrary $Y$ when the maps defining $X$ have algebraic-contraction ratios and there are sufficiently many of them relative to the number of maps defining $Y$. We also show that it holds for a.e. choice of the corresponding contraction ratios, and obtain bounds on the packing dimension of the exceptional parameters. Key ingredients in our argument include a new renormalization procedure in the space of embeddings, and Orponen's projection Theorem for Assouad dimension (2021). Comment: 28 pages |
| Databáze: | arXiv |
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