On the asymptotics of counting functions for Ahlfors regular sets

Dušan Pokorný; Marc Rauch

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 1, page 69-119
  • ISSN: 0010-2628

Abstract

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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 ε → 0 + ε s N ( ε , K ) exist, where K is an s -regular set and N ( ε , K ) is for instance the ε -packing number of K ?

How to cite

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Pokorný, Dušan, and Rauch, Marc. "On the asymptotics of counting functions for Ahlfors regular sets." Commentationes Mathematicae Universitatis Carolinae 62 63.1 (2022): 69-119. <http://eudml.org/doc/299267>.

@article{Pokorný2022,
abstract = {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 $\lim _\{\varepsilon \rightarrow 0+\} \varepsilon ^s N(\varepsilon ,K)$ exist, where $K$ is an $s$-regular set and $N(\varepsilon ,K)$ is for instance the $\varepsilon $-packing number of $K$?},
author = {Pokorný, Dušan, Rauch, Marc},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Ahlfors regular; $s$-regular; packing number; Minkowski measurability; renewal theory},
language = {eng},
number = {1},
pages = {69-119},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the asymptotics of counting functions for Ahlfors regular sets},
url = {http://eudml.org/doc/299267},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Pokorný, Dušan
AU - Rauch, Marc
TI - On the asymptotics of counting functions for Ahlfors regular sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 1
SP - 69
EP - 119
AB - 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 $\lim _{\varepsilon \rightarrow 0+} \varepsilon ^s N(\varepsilon ,K)$ exist, where $K$ is an $s$-regular set and $N(\varepsilon ,K)$ is for instance the $\varepsilon $-packing number of $K$?
LA - eng
KW - Ahlfors regular; $s$-regular; packing number; Minkowski measurability; renewal theory
UR - http://eudml.org/doc/299267
ER -

References

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