Affine embeddings of Cantor sets on the line

TL;DR

This paper investigates conditions under which affine embeddings of Cantor sets imply a logarithmic relation between their contraction ratios, providing evidence for a conjecture linking geometric embeddings and algebraic properties.

Contribution

It proves that near-equal Hausdorff dimensions of Cantor sets imply logarithmic commensurability of contraction ratios when one set embeds into another, advancing the conjecture by Feng, Huang, and Rao.

Findings

Contraction ratios are logarithmically commensurable under certain conditions.

Provides partial evidence for a conjecture relating embeddings to algebraic ratios.

Combines Feng-Huang-Rao approach with Hochman's entropy results.

Abstract

Let $s\in (0,1)$, and let $F\subset \mathbb{R}$ be a self similar set such that $0 < \dim_H F \leq s$ . 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 $0 \leq \dim_H E - \dim_H F < \delta$ then (under some mild conditions on $E$ and $F$) the contraction ratios of $E$ and $F$ are logarithmically commensurable. This provides more evidence for a Conjecture of Feng, Huang, and Rao, 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, with a new result by Hochman, which is related to the increase of entropy of measures under convolutions.