Affine embeddings of Cantor sets on the line

Abstract

Let s \in (0,1) , and let F \subset \mathbb R be a self similar set such that 0 < \mathrm {dim}_H F \leq s . We prove that there exists \delta = \delta (s) > 0 such that if F admits an affine embedding into a homogeneous self similar set E and 0 \leq \mathrm {dim}_H E – \leq \mathrm {dim}_H F < \delta then (under some mild conditions on E and F ) the contraction ratios of E and F are logarithmically commensurable. This provides more evidence for Conjecture 1.2 of Feng, Huang, and Rao [7], that states that these contraction ratios are logarithmically commensurable whenever F admits an affine embedding into E (under some mild conditions). Our method is a combination of an argument based on the approach of Feng, Huang, and Rao in [7] with a new result by Hochman [10], which is related to the increase of entropy of measures under convolutions.