How much can heavy lines cover?

Abstract

One formulation of Marstrand's slicing theorem is the following. Assume that , and is a Borel set with . Then, for almost all directions , almost all of is covered by lines parallel to with . We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction , is it true that a strictly less than ‐dimensional part of is covered by the heavy lines , namely those with ? A positive answer for ‐regular sets was previously obtained by the first author. The answer for general Borel sets turns out to be negative for and positive for . More precisely, the heavy lines can cover up to a dimensional part of in a generic direction. We also consider the part of covered by the ‐heavy lines, namely those with for . We establish a sharp answer to the question: how much can the ‐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.