# On the asymptotics of counting functions for Ahlfors regular sets

## TL;DR

This paper investigates the asymptotic behavior of counting functions for Ahlfors regular sets in metric spaces, linking them to tree-like structures and analyzing the existence of certain limits related to packing numbers.

## Contribution

It establishes a connection between Ahlfors regular sets and tree-like structures and studies conditions for the existence of limits of scaled counting functions.

## Key findings

- Ahlfors regular sets correspond to specific tree-like structures.
- Conditions for the existence of the limit of ε^s N(ε,K) are identified.
- Insights into the asymptotic properties of packing numbers for these sets.

## Abstract

In this paper we deal with the so-called Ahlfors regular sets (also known as $s$-regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: under which conditions does the limit $\lim_{\varepsilon\to 0+} \varepsilon^s N(\varepsilon,K)$ exist, where $K$ is an $s$-regular set and $N(\varepsilon,K)$ is for instance the $\varepsilon$-packing number of $K$?