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 { ϕ i (x) = β x + i 1 - β N : i = 0 , 1 , … , N }. <graphic href="209_2024_3499_Article_Equ27.gif"></graphic> For m ∈ Z + we show that there exist infinitely many translation vectors t = (t 0 , t 1 , … , t m) with 0 = t 0 < t 1 < ⋯ < t m such that the union ⋃ j = 0 m (Γ + t j) is a self-similar set. Furthermore, for 0 < β < 1 / (2 N + 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. [ABSTRACT FROM AUTHOR]