Multiplication on self-similar sets with overlaps

Let $A,B\subset \mathbb{R}$. Define $$A\cdot B=\{x\cdot y:x\in A,y\in B\}\mathrm{.}$$ 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$I=[0,1]$istheconvexhullof$K$\mathrm{.}Themainresultofthispaperis$K\cdot K=[0,1]$ if and only if $(1-\lambda {)}^2\le c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in [0,1]$, $u=x\cdot y$, where $x,y\in K$.