Dimension conservation for self-similar sets and fractal percolation

TL;DR

This paper develops a projection-based technique to establish dimension conservation properties for sections of self-similar and fractal percolation sets, revealing near-maximal dimension of sections in most directions.

Contribution

It introduces a novel method linking projection properties to dimension conservation for self-similar and percolation fractals, extending understanding of their geometric structure.

Findings

Dimension conservation holds for most directions in self-similar sets with full rotation group.

Sections of Mandelbrot percolation sets have similar dimension inequalities.

For certain self-similar sets, all directions satisfy the dimension bounds with box dimension.

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.