Multiplication on self-similar sets with overlaps

Let $A,B\subset\mathbb{R}$. Define $$A\cdot B=\{x\cdot y:x\in A, y\in B\}.$$ In this paper, we consider the following class of self-similar sets with overlaps. Let $K$ be the attractor of the IFS $\{f_1(x)=\lambda x, f_2(x)=\lambda x+c-\lambda,f_3(x)=\lambda x+1-\lambda\}$, where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main result of this paper is $K\cdot K=[0,1]$ if and only if $(1-\lambda)^2\leq c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in[0,1]$, $u=x\cdot y$, where $x,y\in K$.