Fat fractal percolation and k-fractal percolation

We consider two variations on theMandelbrot fractal percolationmodel. In the k-fractal percolation model, the d-dimensional unit cube is divided in Nd equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N→∞ This is analogous to the result of Falconer and Grimmett (1992) that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability pn at step n of the construction, where (p_n)_n≥ is a non-decreasing sequence with π^∞_n=1 p_n > 0. The Lebesgue measure of the limit set is positive a.s. given nonextinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s.