On the Hausdorff measure of shrinking target sets on self-conformal sets

TL;DR

This paper characterizes the Hausdorff measure of shrinking target sets on self-conformal sets, establishing a dichotomy based on the convergence or divergence of a sum related to the target radii.

Contribution

It provides a complete characterization of the Hausdorff measure for these sets, extending previous results on their Hausdorff dimension.

Findings

Dichotomy in Hausdorff measure depending on sum convergence/divergence

Complements earlier work on Hausdorff dimension of these sets

Advances understanding of measure-theoretic properties of shrinking targets

Abstract

In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets". Our main result complements earlier work of Levesley, Salp, and Velani~(2007), and recent work of Baker (2019).