On the union of homogeneous symmetric Cantor set with its translations

Fix a positive integer N and a real number 0<β<1/(N+1). Let Γ be the homogeneous symmetric Cantor set generated by the IFS (Formula presented.) For m∈Z₊ we show that there exist infinitely many translation vectors t=(t₀,t₁,…,t_m) with 0=t₀<t₁<⋯<t_m such that the union ⋃_j=0^m(Γ+t_j) is a self-similar set. Furthermore, for 0<β<1/(2N+1), we give a finite algorithm to determine whether the union ⋃_j=0^m(Γ+t_j) is a self-similar set for any given vector t. Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.