On the union of homogeneous symmetric Cantor set with its translations

Abstract: Fix a positive integer $N$ and a real number $0< \beta < 1/(N+1)$. Let $\Gamma$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\cdots, N \Big}. $$ For $m\in\mathbb{Z}+$ we show that there exist infinitely many translation vectors $\mathbf t=(t_0,t_1,\cdots, t_m)$ with $0=t_0<t_1<\cdots<t_m$ such that the union $\bigcup{j=0}^m(\Gamma+t_j)$ is a self-similar set. Furthermore, for $0< \beta < 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(\Gamma+t_j)$ is a self-similar set. Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.