Arithmetic products of Cantor sets.

Question

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?

Answer 1

The results my paper with M. Hochman yield the following result on the products of self-similar sets:

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Let $A=\bigcup_{i=1}^m r_i A + t_i$, $B=\bigcup_{i=1}^n s_i B+u_i$ be two self-similar 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.

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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 self-similar, 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.

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One can also say something in the case when the irrationality condition does not hold, for example if $A,B$ are homogeneous self-similar sets (all the contraction ratios are equal), with the same contraction ratio. In this case, $E=A\times B$ is a self-similar 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 self-similar 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 self-similar 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.

Answer 2

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.