Self-similar structure on the intersection of middle-(1 - 2β) Cantor sets with β ∈ (1/3, 1/2)

Let Γ_β be the middle-(1 - 2β) Cantor set with β ∈ (1/3, 1/2). We give all real numbers t with unique {-1, 0, 1}-code such that the intersections Γ_β ∩ (Γ _β+t) are self-similar sets. For a given β ∈ (1/3, 1/2), a criterion is obtained to check whether or not a t ∈ [-1, 1] has the unique {-1, 0, 1}-code from both geometric and algebraic views.