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- W2950796651 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 $f_1(I)cap f_2(I)neq emptyset, (f_1(I)cup f_2(I))cap f_3(I)=emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main results of this paper are as follows: $$sqrt{K}+sqrt{K}=[0,2]$$ if and only if $$sqrt{c}+1geq 2sqrt{1-lambda},$$ where $sqrt{K}+sqrt{K}={sqrt{x}+sqrt{y}:x,yin K}$. If $cgeq (1-lambda)^2$, then $$dfrac{K}{K}=left{dfrac{x}{y}:x,yin K, yneq 0right}=left[0,inftyright).$$ As a consequence, we prove that the following conditions are equivalent: (1) For any $uin [0,1]$, there are some $x,yin K$ such that $u=xcdot y;$ (2) For any $uin [0,1]$, 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 $$u=x_1+x_2=x_3-x_4=x_5cdot x_6=x_7div x_8=sqrt{x_9}+sqrt{x_{10}};$$ (3) $cgeq (1-lambda)^2$." @default.
- W2950796651 created "2019-06-27" @default.
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- W2950796651 date "2018-10-11" @default.
- W2950796651 modified "2023-09-26" @default.
- W2950796651 title "Multiple representations of real numbers on self-similar sets with overlaps" @default.
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- W2950796651 doi "https://doi.org/10.48550/arxiv.1810.04930" @default.
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