Interior points and Lebesgue measure of overlapping Mandelbrot percolation sets

We consider a special one-parameter family of d-dimensional random, homogeneous self-similar iterated function systems (IFSs) satisfying the finite type condition. The object of our study is the positivity of Lebesgue measure and the existence of interior points in these random sets and in particular, the existence of an interesting parameter interval where the attractor has positive Lebesgue measure but empty interior, almost surely conditioned on the attractor not being empty. We give a sharp bound on the critical probability for the case of positive Lebesgue measure using the theory of multitype branching processes in random environments. Further, in some special cases, we give a bound on the critical probability for the existence of interior points. Using a recent result of Tom Rush, we also provide a family of such random sets where there exists a parameter interval for which the corresponding attractor has a positive Lebesgue measure, but empty interior almost surely conditioned on the attractor not being empty.