On the union of homogeneous symmetric Cantor set with its translations

Fix a positive integer N and a real number \(0< \beta < 1/(N+1)\). Let \(\Gamma \) be the homogeneous symmetric Cantor set generated by the IFS

$$\begin{aligned} \Bigg \{ \phi _i(x)=\beta x + i \frac{1-\beta }{N}: i=0,1,\ldots , N \Bigg \}. \end{aligned}$$

For \(m\in \mathbb {Z}_+\) we show that there exist infinitely many translation vectors \({\textbf{t}}=(t_0,t_1,\ldots , t_m)\) 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 finite algorithm to determine whether the union \(\bigcup _{j=0}^m(\Gamma +t_j)\) is a self-similar set for any given vector \({\textbf{t}}\). Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.