Description
The Mandelbrot percolation fractal in the plane is constructed inductively as follows. We fix an integer K>1 and a probability 0<p≤1. The closed unit square is divided into K^2 congruent sub-squares, each of which is independently retained with probability p and discarded with probability 1-p. This process is repeated in the retained cubes ad infinitum, or until there are no cubes left. The Mandelbrot percolation process restricted to the building blocks of a sponge (for instance, the Sierpinski carpet) yields a random sponge.
In this talk, we study the positivity of Lebesgue measure of rational projections of random sponges onto lines. Compared with the case of Mandelbrot percolation, spatial inhomogeneity of the sponge yields qualitatively different phenomena.
This is based on joint work with Károly Simon.