New results on embeddings of self‐similar sets via renormalization
Abstract
Abstract For self‐similar sets , we obtain new results toward 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 have algebraic contraction ratios, and also for arbitrary when the maps defining have algebraic contraction ratios and there are sufficiently many of them relative to the number of maps defining . We also show that it holds for almost every 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).
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Proceedings of the London Mathematical Society
- Publisher
- Wiley
- Volume
- 133
- Issue
- 1
- Publication date
- 1 July 2026
- Language
- EN