How much can heavy lines cover?

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