How much can heavy lines cover?

Abstract

One formulation of Marstrand’s slicing theorem is the following. Assume that𝑡∈(1,2],and𝐵⊂ℝ2is a Borel set with 𝑡(𝐵)<∞. Then, for almost all directions𝑒∈𝑆1, 𝑡 almost all of 𝐵 is covered by lines𝓁parallel to 𝑒 with dim H(𝐵∩𝓁)=𝑡−1. We investigate the prospects of sharpening Marstrand’s result in the following sense: in a generic direction𝑒∈𝑆1, is it true that a strictly less than 𝑡-dimensional part of𝐵is covered by the heavy lines𝓁⊂ℝ2, namely those with dim H(𝐵∩𝓁)>𝑡−1? A positive answer for𝑡-regular sets𝐵⊂ℝ2was previously obtained by the first author. The answer for general Borel sets turns out to be negative for𝑡∈(1,32] and positive for𝑡∈(32,2]. More precisely, the heavy lines can cover up to amin{𝑡,3−𝑡} dimensional part of𝐵in a generic direction. We also consider the part of𝐵covered by the𝑠-heavy lines, namely those with dim H(𝐵∩𝓁)⩾𝑠for𝑠>𝑡−1. We establish a sharp answer to the question: how much can the𝑠-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.