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 dimHE≤1, for fractal
percolations (case dimHE>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).
(This will be roughly a repetition of M.Rams talk at University of
Warsaw dynamical systems seminar on Friday.)