Abstract: Let C be the Cantor ternary set. The condition of the intersection C\cap (C+t)\cap (C+s)\neq\emptyset was considered and the Hausdorff dimension of (t,s) was computed when the intersection was nonempty. A conclusion was proved: dim_H C\cap (C+t)\cap (C+s)=0 for a.e. (t,s)\in{\bf R}\times {\bf R}. Then by a related result of Moran set, the expression of dim_H C\cap (C+t)\cap (C+s) was investigated.

Key words: intersection, end-points, Moran sets, Cantor ternary set, intersection, end-points, Moran sets