Fractal dimension and measure of the subset of Moran set

We discuss the fractal dimension and measure for the subset BP(ω) of Moran set E(ω) in Rd satisfying the strong separation condition. Firstly, we give the Hausdorff dimension of subset BP(ω) in compatible case and incompatible case. Then we attain that there exists a subset B of the set BP(ω) such that B has full μP-measure but zero Hausdorff measure in incompatible case. Finally, if the gap condition holds, we see that BP(ω) and E(ω) have the same Hausdorff measure and packing measure, and both of them are α-sets in compatible case.