# Affine embeddings of Cantor sets on the line

### Amir Algom

The Hebrew University of Jerusalem, Israel

## Abstract

Let $s∈(0,1)$, and let $F⊂R$ be a self similar set such that $0<dim_{H}F≤s$. We prove that there exists $δ=δ(s)>0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$ and $0≤dim_{H}E–≤dim_{H}F<δ$ 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.

## Cite this article

Amir Algom, Affine embeddings of Cantor sets on the line. J. Fractal Geom. 5 (2018), no. 4, pp. 339–350

DOI 10.4171/JFG/63