Projections of fractal percolations

Michal Rams (IM PAN Varsóvia)

The Marstrand Theorem is one of (possibly, the) most important results in geometric measure theory. It states that for any set X 2 R2 of Hausdorff dimension s for almost all  2 P1 the projections (X) (where  is an orthogonal projection to a line in direction ) have Hausdorff dimension s (if s  1) or have positive Lebesgue measure (if s > 1). Many generalizations exist (higher dimensional versions, estimations on the size of the set of exceptional projections, nonlinear versions etc.). The result I will present, joint with Karoly Simon from Budapest Technical University, is as follows. We consider a naturally defined set-valued random variable in R2, so called fractal percolation. We prove that almost every realization of fractal percolation satisfies the assertion of Marstrand Theorem for all (not almost all) directions. Moreover, in case s > 1 not only every projection has positive Lebesgue measure, it even contains an interval, The statement can be generalized for some other classes of projections. For example, if X is a realization of fractal percolation of Hausdorff
dimension s > 1 then for every x 2 R2 the set of angles under which X is visible from x contains an interval.