  Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we
show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely
embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$,
then the Hausdorff dimension of the intersection $E\cap f(F)$ is strictly less
than that of $F$ for any $C^1$-diffeomorphism $f$ on $\R^d$. Under certain
circumstances, we prove the logarithmic commensurability between the
contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As
an application, we show that $\dim_HE\cap f(F)<\min\{\dim_HE, \dim_HF\}$ when
$E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\geq 2$ are two
integers with $\log p/\log q\not \in \Q$. This is related to a conjecture of
Furtenberg about the intersections of Cantor sets.
