Hausdorff dimension, intersection of projections and exceptional plane sections

TL;DR

This paper investigates the dimensions of projections and intersections of fractal sets in the plane, establishing new sharp bounds and showing that exceptions to classical projection theorems are rare in a Hausdorff dimension sense.

Contribution

It provides new results on the dimensions of projections and intersections of fractal sets, including sharp bounds and rarity of exceptions in the plane.

Findings

Projections of sets with certain dimension conditions intersect with dimension close to the minimum of the set dimensions.

Constructed examples show the sharpness of the bounds for intersection dimensions.

The set of points where Marstrand's projection theorem fails has Hausdorff dimension at most one.

Abstract

This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion to the planar cases of our theorems. Our first main result considers the orthogonal projections of two Borel sets $A,B \subset \mathbb{R}^{2}$ into one-dimensional subspaces. Under the assumptions $\dim A \leq 1 < \dim B$ and $\dim A + \dim B > 2$, we prove that the intersection of the projections $P_{L}(A)$ and $P_{L}(B)$ has dimension at least $\dim A - \epsilon$ for positively many lines $L$, and for any $\epsilon > 0$. This is quite sharp: given $s,t \in [0,2]$ with $s + t = 2$, we construct compact sets $A,B \subset \mathbb{R}^{2}$ with $\dim A = s$ and $\dim B = t$ such that almost all intersections $P_{L}(A) \cap P_{L}(B)$ are empty. In case both $\dim A > 1$ and $\dim B > 1$, we prove that the intersections $P_{L}(A) \cap P_{L}(B)$ have positive length for positively many $L$. If $A \subset \mathbb{R}^{2}$ is a Borel set with $0 < \mathcal{H}^{s}(A) < \infty$ for some $s > 1$, it is known that $A$ is 'visible' from almost all points $x \in \mathbb{R}^{2}$ in the sense that $A$ intersects a positive fraction of all lines passing through $x$. In fact, a result of Marstrand says that such non-empty intersections typically have dimension $s - 1$. Our second main result strengthens this by showing that the set of exceptional points $x \in \mathbb{R}^{2}$, for which Marstrand's assertion fails, has Hausdorff dimension at most one.