A New Hausdorff Content Bound for Limsup Sets

TL;DR

This paper introduces a new Hausdorff content bound for limsup sets, extending the mass transference principle and applying to various problems in metric spaces, with simplified proofs and broader applicability.

Contribution

It presents a novel Hausdorff content bound for limsup sets that generalizes existing principles and applies to all complete metric spaces without regularity assumptions.

Findings

New proof and generalization of the mass transference principle

Bounds on random limsup sets without Ahlfors regularity

Simplified arguments for Hausdorff content estimation

Abstract

We give a new Hausdorff content bound for limsup sets, which is related to Falconer's sets of large intersection. Falconer's sets of large intersection satisfy a content bound for all balls in a space. In comparison, our main theorem only assumes a scale-invariant bound for the balls forming the limit superior set in question. We give four applications of these ideas and our main theorem: a new proof and generalization of the mass transference principle related to Diophantine approximations, a related result on random limsup sets, a new proof of Federer's characterization of sets of finite perimeter and a statement concerning generic paths and the measure theoretic boundary. The new general mass transference principle transfers a content bound of one collection of balls, to the content bound of another collection of sets -- however, this content bound must hold on all balls in the space. The benefit of our approach is greatly simplified arguments as well as new tools to estimate Hausdorff content. The new methods allow for us to dispense with many of the assumptions in prior work. Specifically, our general Mass Transference Principle, and bounds on random limsup sets, do not assume Ahlfors regularity. Further, they apply to any complete metric space. This generality is made possible by the fact that our general Hausdorff content estimate applies to limsup sets in any complete metric space.