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

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