Multiplication on self-similar sets with overlaps

Abstract

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 $f_1(I)\cap f_2(I)\ne \varnothing ,(f_1(I)\cup f_2(I))\cap f_3(I)=\varnothing ,$ 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 $(1-\lambda {)}^2\le c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in [0,1]$, $u=x\cdot y$, where $x,y\in K$.

12 pages