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 $0<{\dim }_{H}F\le s$. 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 $0\le {\dim }_{H}E–\le {\dim }_{H}F<\delta$ 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.