Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product $AB=\lbrace aba\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are selfsimilar?
2 Answers
The results my paper with M. Hochman yield the following result on the products of selfsimilar sets:
Let $A=\bigcup_{i=1}^m r_i A + t_i$, $B=\bigcup_{i=1}^n s_i B+u_i$ be two selfsimilar sets on $\mathbb{R}$. Suppose that $\log r_i/\log s_j$ is irrational for some $i,j$. Then $$ \dim_H(A\cdot B)=\min(\dim_H(A)+\dim_H(B),1), $$ where $\dim_H$ stands for Hausdorff dimension.
The idea is as follows: by replacing $A$ by $A$ and $B$ by $B$ if needed, we may assume that both $A\cap (0,\infty)$ and $B\cap (0,\infty)$ contain a rescaled copy of $A,B$ respectively. It is enough to prove the result for these rescaled copies, so we may in fact assume that $A,B\subset (0,\infty)$.
Now the sets $A'=\{ \log a:a\in A\}$ and $B'=\{ \log b:b\in B\}$ are not quite selfsimilar, but they are the attractor of a smooth IFS which are conjugated to the original ones via the logarithm map, so locally they look the same (and have the same Hausdorff dimensions). By Corollary 1.5 in our paper, $$ \dim_H(A'+B') = \min(\dim_H(A)+\dim_H(B),1), $$ but since $A\cdot B$ is obtained from $A'+B'$ by exponentiation, it has the same dimension and we are done.
One can also say something in the case when the irrationality condition does not hold, for example if $A,B$ are homogeneous selfsimilar sets (all the contraction ratios are equal), with the same contraction ratio. In this case, $E=A\times B$ is a selfsimilar set in $\mathbb{R}^2$, and M. Hochman has recently proved a very impressive result saying that if $P_\theta$ is the orthogonal projection with angle $\theta$, then $$ \dim_H(P_\theta E) = \min(\dim_H(A)+\dim_H(B),1) $$ outisde of a set of $\theta$ of Hausdorff (and packing) dimension $0$, and the same is true for selfsimilar measures on $E$. Even though this result is for affine maps and here we are interested in the nonlinear map $P(x,y)=x.y$, combined with Theorem 1.23 in this other paper by M. Hochman, and the fact that selfsimilar measures are homogeneous in the sense of that paper, we get that also $$ \dim_H(A\cdot B) = \\min(\dim_H(A)+\dim_H(B),1). $$
It would be interesting to know if $A\cdot B$ has positive measure when the sum of the dimensions exceeds $1$, but this seems currently out of reach.

$\begingroup$ Many thanks for this detailed answer. I agree: It would be very interesting to know if AB has positive length, under some condition. Very Best 9i $\endgroup$ Commented May 31, 2013 at 21:04
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.