The local dimensions of some Moran measures with open set condition

Dai and Liu obtained the formula of local dimensions of some Moran measures on Moran sets in RdRd under the strong separation condition. In this paper, we prove that the result is still true under the open set condition. Due to the lack of the strong separation condition, our approach is essentially different to that used by Dai and Liu. We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples.