Affine embeddings of Cantor sets and dimension of $\alpha\beta$-sets

TL;DR

This paper investigates the conditions under which one self-similar set can be affinely embedded into another, revealing a relationship between their contraction ratios, and studies related multi-rotation invariant sets on the circle.

Contribution

It establishes logarithmic commensurability of contraction ratios for dust-like self-similar sets with small Hausdorff dimension and analyzes the box-counting dimension of multi-rotation invariant sets.

Findings

Proves logarithmic commensurability between contraction ratios of embedded sets.

Provides partial confirmation of a conjecture in fractal geometry.

Analyzes the box-counting dimension of $eta$-sets and similar structures.

Abstract

Let $E, F\subset {\Bbb R}^d$ be two self-similar sets, and suppose that $F$ can be affinely embedded into $E$. Under the assumption that $E$ is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$. This gives a partial affirmative answer to Conjecture 1.2 in \cite{FHR14}. The proof is based on our study of the box-counting dimension of a class of multi-rotation invariant sets on the unit circle, including the $\alpha\beta$-sets initially studied by Engelking and Katznelson.