  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.
