| This paper focus on the structure of fractals,including two topics.The first is Ahlfors-David regular subsets of fractal sets.The second is the arithmetic sum of Cantor sets.Ahlfors-David regularity implies a uniform structure of measure and geometric.We study the compact Ahlfors-David regular subsets of fractals.For this,we introduce the Ahlfors-David regular dimension and obtain some basic properties of this dimension.Based on these properties and some known results in dimension theory of fractals,we compute Ahlfors-David regular dimensions for Moran sets and Mc Mullen sets.We also establish connections between Ahlfors-David regular dimension,Assouad dimension and lower Assouad dimension.In the study of Moran sets in R,we use an technique of separation to solve the difficulty of boundary interference,and derive the upper bound of its lower Assouad dimension.In the study of Mc Mullen sets,we show that the dimension of its Ahlfors-David regular subset can be arbitrarily close to its Hausdorff dimension.It is a new way to caculate the lower bound of Hausdorff dimension and shows that the Mc Mullen set is regular in some sense.By a “Cantor set” we mean a compact perfect totally disconnected subsets of R.We study the problem that what kind of geometric structure of a Cantor set ensures the finite arithmetic sums of it being a closed interval.Based on the connection between the Newhouse thickness,uniformly perfect space and quasisymmetric embedding,we obtain that a uniformly perfect Cantor set can generate a closed interval by finite arithmetic sums.For a kind of family of sets without the uniformly perfect property,we introduce the definition of piecewise complementary of uniformly perfection,which is related to some known results.Newhouse thickness theory cannot give the minimum times of arithmetic sum for generating closed intervals,and it is difficult to find a simple formula to do so.For a certain kind of self-similar sets,we present an algorithm for calculating the minimum number. |
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