How much can heavy lines cover?

Abstract

One formulation of Marstrand's slicing theorem is the following. Assume that $t \in (1,2]$, and $B \subset \mathbb{R}^{2}$ is a Borel set with $\mathcal{H}^{t}(B) < \infty$. Then, for almost all directions $e \in S^{1}$, $\mathcal{H}^{t}$ almost all of $B$ is covered by lines $\ell$ parallel to $e$ with $\dim_{\mathrm{H}} (B \cap \ell) = t - 1$. We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction $e \in S^{1}$, is it true that a strictly less than $t$-dimensional part of $B$ is covered by the heavy lines $\ell \subset \mathbb{R}^{2}$, namely those with $\dim_{\mathrm{H}} (B \cap \ell) > t - 1$? A positive answer for $t$-regular sets $B \subset \mathbb{R}^{2}$ was previously obtained by the first author. The answer for general Borel sets turns out to be negative for $t \in (1,\tfrac{3}{2}]$ and positive for $t \in (\tfrac{3}{2},2]$. More precisely, the heavy lines can cover up to a $\min\{t,3 - t\}$ dimensional part of $B$ in a generic direction. We also consider the part of $B$ covered by the $s$-heavy lines, namely those with $\dim_{\mathrm{H}} (B \cap \ell) \geq s$ for $s > t - 1$. We establish a sharp answer to the question: how much can the $s$-heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors-regular sets to the class of sub-uniformly distributed sets.