Projections of fractal percolations

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.