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Keywords:
Ahlfors regular; $s$-regular; packing number; Minkowski measurability; renewal theory
Ahlfors regular; $s$-regular; packing number; Minkowski measurability; renewal theory
Summary:
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 $K$?
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