Some Hausdorff Dimensional Results for Homogeneous Moran Sets in $${\mathbb {R}}^d$$ R d

Abstract

In this paper, we construct a class of special homogeneous Moran set in $${\mathbb {R}}^d(d\ge 1)$$ , called $$\{m_{k}^d\}$$ -quasi homogeneous Cantor set and obtain its Hausdorff dimension. By adjusting the value of $$\{m_{k}^d\}_{k\ge 1}$$ , we constructively prove the intermediate value theorem for the homogeneous Moran set in $${\mathbb {R}}^d$$ . Moreover, we obtain a sufficient condition for the Hausdorff dimensions of homogeneous Moran sets in $${\mathbb {R}}^d$$ to assume the minimum value, which expands earlier works.