The Dimension of Projections of Fractal Percolations

Fractal percolation or Mandelbrot percolation is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $Esubsetmathbb{R}^{2}$ a typical realization of the fractal percolation on the plane,

• If $dim_{rm H}E<1$ then for all lines ? the orthogonal projection E _? of E to ? has the same Hausdorff dimension as E,
• If $dim_{rm H}E>1$ then for any smooth real valued function f which is strictly increasing in both coordinates, the image f(E) contains an interval.

The second statement is quite interesting considering the fact that E is almost surely a Cantor set (a random dust) for a large part of the parameter domain, see Chayes et al. (Probab. Theory Relat. Fields, 77(3):307–324, 1988). Finally, we solve a related problem about the existence of an interval in the algebraic sum of d≥2 one-dimensional fractal percolations.