Dimension (In)equalities and Holder Continuous Curves in Fractal Percolation

We relate various concepts of fractal dimension of the limiting set in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non-trivial Holder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1