Multiple representations of real numbers on self-similar sets with overlaps

Abstract

Let $K$ be the attractor of the following 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 results of this paper are as follows: $$\sqrt{K}+\sqrt{K}=[0,2]$$ if and only if $$\sqrt{c}+1\ge 2\sqrt{1-\lambda },$$ where $\sqrt{K}+\sqrt{K}=\{\sqrt{x}+\sqrt{y}:x,y\in K\}$. If $c\ge (1-\lambda {)}^2$, then $$\frac{K}{K}=\left\{\frac{x}{y}:x,y\in K,y\ne 0\right\}=\left[0,\infty \right).$$ As a consequence, we prove that the following conditions are equivalent: (1) For any $u\in [0,1]$, there are some $x,y\in K$ such that $u=x\cdot y;$ (2) For any $u\in [0,1]$, there are some $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10}\in K$ such that $$u=x_1+x_2=x_3-x_4=x_5\cdot x_6=x_7÷x_8=\sqrt{x_9}+\sqrt{x_{10}};$$ (3) $c\ge (1-\lambda {)}^2$.

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