PROJECTIONS OF THE RANDOM MENGER SPONGE

In this paper we prove theorems about a special family of random self-similar sets on the line, we apply these theorems to get the Hausdorff dimension, the Lebesgue measure and existence of interior points of some projections of the random right-angled Sierpiski gasket, the random Sierpinski carpet and the random Menger sponge. The Menger sponge is one of the most well-known example of self-similar sets in R-3. The Mandelbrot percolation process restricted to the cubes, which are the building blocks of the Menger sponge, yields the random Menger sponge, a random self-similar fractal in R-3. We examine its orthogonal projections to straight lines, from the point of Lebesgue measure and existence of interior points. In particular this yields random self-similar sets on the line with positive Lebesgue measure and empty interior. Moreover, we give a sharp threshold for the probability above which the projections of the random Menger sponge contains an interval in all directions.