Dimension conservation for self-similar sets and fractal percolation

Abstract

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let $K$ be a self-similar subset of ${\mathbb{R}}^2$ with Hausdorff dimension ${\dim }_{H}K>1$ such that the rotational components of the underlying similarities generate the full rotation group. Then for all $\epsilon >0$, writing ${\pi }_{\theta }$ for projection onto the line $L_{\theta }$ in direction $\theta$, the Hausdorff dimensions of the sections satisfy ${\dim }_H(K\cap {\pi }_{\theta }^{-1}x)>{\dim }_{H}K-1-\epsilon$ for a set of $x\in L_{\theta }$ of positive Lebesgue measure, for all directions $\theta$ except for those in a set of Hausdorff dimension 0. For a class of self-similar sets we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.

22 pages, 4 figures