Sections of fractal percolations

I will speak about one recently proved (with Karoly Simon) result about Mandelbrot percolations. We consider a percolation with (almost surely) dimension smaller than 1 and calculate an upper bound for the number of n-th level cylinders that can be intersected by an arbitrary line on the plane. I will then apply this result in two ways. One application, natural, is for generalisation of Marstrand theorem, case dim_H E ≤ 1, for fractal percolations (case dim_H E > 1 was done in one of our previous papers). Another application is for the problem of existence of intervals in the algebraic sum of three random Cantor sets (the paper of Dekking and Simon dealt with sums of two sets; sums of three sets are more difficult because some events become dependent, which makes it difficult to apply tools from theory of large deviations).