The Argument On The Intersectional Characteristics Of High-Dimensional Cantor-Type Set

The fractal set, in this paper, with its characteristics by which the definition comes from, will be discussed. Therefore, various definitions of fractal dimensions, measure and their corresponding property are to be expounded. The relations between measure and dimension as well as that of dimension and dimension will be proved. With that discussion, the results can be used to depict the degree of "thickness" for a fractal set, provide a basis for understanding and applying fractal dimension, and also, clarify certain misled knowledge about fractal dimension. Then, that a fractal set may have inequable values under different definitions to dimensions, and the inequable values of dimensions character different attributes of the fractal set, will be concluded. Secondly, the character of two Cantor-type intersections will be argued on. Let E be a Cantor-type set in higherdimension and let E_α= E +α= {β+α:β∈E}, forα∈[- 1,1]~n. Through the study ofthe fractal structure of E∩E_α, determined are both the relations of the Hausdorffmeasure and dimension between two Cantor-type intersectional sets in higher dimension under given conditions, and obtained is the better bound of the Hausdorff measure.