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 ${\mathcal{H}}^t(B)<\mathrm{\infty }$. 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 $di{m}_H(B\cap \ell )=t-1$. 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 $di{m}_H(B\cap \ell )>t-1$? 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 $t\in (1,\frac{3}{2}]$ and positive for $t\in (\frac{3}{2},2]$. 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 $di{m}_H(B\cap \ell )\ge s$ 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.