Self-similar structure on intersection of Cartesian product of Cantor triadic sets with their translations

Let C be the classical Cantor triadic set. For a,b ? [-1,1]{alpha,betain [-1,1]} , a sufficient and necessary condition for (C×C)?(C×C+(a,b)){(Ctimes C)cap (Ctimes C+(alpha,beta))} to be self-similar is obtained.