TITLE: Arithmetic products of Cantor sets.
QUESTION [7 upvotes]: Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product
$AB=\lbrace ab|a\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are self-similar?

REPLY [2 votes]: I) If $K$ and $K'$ are arbitrary Cantor sets with
dim$_HK=\overline{\dim}_BK$ and $HD (K)+HD(K')<1$,   then  $\dfrac{K}{ K'}$ and  $K\cdot K'$ have zero Lebesgue measure.
II) If $\tau(K)\cdot\tau(K')>1$, then  $\dfrac{K}{ K'}$ and  $K\cdot K'$ contain an interval.
III) Let  $C_\alpha$ and $C_\beta$ be two middle Cantor sets with  $\dfrac{\log \alpha}{\log \beta}=\dfrac{n_0}{m_0},~(m_0,~n_0)=1$ and $\dfrac{1}{\gamma}<\tau(C_\alpha)\cdot\tau(C_\beta)\leq1$, where  $\gamma:=\alpha^{-\dfrac{1}{n_0}}$. 
Then $\dfrac{C_\alpha}{ C_\beta}$ and $C_\alpha \cdot  C_\beta$  contain  an interval.
See the proofs in section 3 of my article on arXiv.