On the union of homogeneous symmetric Cantor set with its translations

Fix a positive integer N and a real number \(0< \beta < 1/(N+1)\) $0<\beta <1/(N+1)$ . Let \(\Gamma \) $\mathrm{\Gamma }$ be the homogeneous symmetric Cantor set generated by the IFS \(\begin{aligned} \Bigg \{ \phi _i(x)=\beta x + i \frac{1-\beta }{N}: i=0,1,\ldots , N \Bigg \}. \end{aligned}\) $$\begin{array}{c}\hfill \{{\varphi }_i(x)=\beta x+i\frac{1-\beta }{N}:i=0,1,\dots ,N\}.\end{array}$$ For \(m\in \mathbb {Z}_+\) $m\in {\mathbb{Z}}_{+}$ we show that there exist infinitely many translation vectors \({\textbf{t}}=(t_0,t_1,\ldots , t_m)\) $\mathbf{t}=(t_0,t_1,\dots ,t_m)$ with \(0=t_0<t_1<\cdots <t_m\) $0=t_0<t_1<\cdots <t_m$ such that the union \(\bigcup _{j=0}^m(\Gamma +t_j)\) ${\bigcup }_{j=0}^m(\mathrm{\Gamma }+t_j)$ is a self-similar set. Furthermore, for \(0< \beta < 1/(2N+1)\) $0<\beta <1/(2N+1)$ , we give a finite algorithm to determine whether the union \(\bigcup _{j=0}^m(\Gamma +t_j)\) ${\bigcup }_{j=0}^m(\mathrm{\Gamma }+t_j)$ is a self-similar set for any given vector \({\textbf{t}}\) $\mathbf{t}$ . Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.