In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{ R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than 1 then the orthogonal projection to \textbf{ every} line, the radial projection with \textbf{ every} center, and distance set from \textbf{ every} point contain intervals.
Categories: math.DS math.PR