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 $| for projection onto the |$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.