Paper detail

Dimension conservation for self-similar sets and fractal percolation

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 $ε>0$, writing $π_θ$ for projection onto the line $L_θ$ in direction $θ$, the Hausdorff dimensions of the sections satisfy $\dim_H (K\cap π_θ^{-1}x)> \dim_H K - 1 - ε$ for a set of $x \in L_θ$ of positive Lebesgue measure, for all directions $θ$ 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.