On the union of homogeneous symmetric Cantor set with its translations

Fix a positive integer $N$ and a real number $0<\beta <1\mathrm{/}(N+1)$. Let $\mathrm{\Gamma }$ be the homogeneous symmetric Cantor set generated by the IFS $$\{{\phi }_i(x)=\beta x+i\frac{1-\beta }{N}:i=0,1,\cdots ,N\}\mathrm{.}$$ 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 ${\bigcup }_{j=0}^m(\mathrm{\Gamma }+t_j)$ is a self-similar set. Furthermore, for $0<\beta <1\mathrm{/}(2N+1)$, we give a complete characterization on which the union ${\bigcup }_{j=0}^m(\mathrm{\Gamma }+t_j)$ 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.