Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings

Marc Kesseböhmer; Sabrina Kombrink

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arXiv · 1702.02854

Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings

Marc Kesseböhmer·Sabrina Kombrink

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Abstract

We give conditions for the existence of the Minkowski content of limit sets stemming from infinite conformal graph directed systems. As an application we obtain Minkowski measurability of Apollonian gaskets, provide explicit formulae of the Minkowski content, and prove the analytic dependence on the initial circles. Further, we are able to link the fractal Euler characteristic, as well as the Minkowski content, of Apollonian gaskets with the asymptotic behaviour of the circle counting function studied by Kontorovich and Oh. These results lead to a new interpretation and an alternative formula for the Apollonian constant. We estimate a first lower bound for the Apollonian constant, namely $0{.}055$, partially answering an open problem by Oh of 2013. In the higher dimensional setting of collections of disjoint balls, generated e.\,g.\ by Kleinian groups of Schottky type, we prove that all fractal curvature measures exist and are constant multiples of each other. Further number theoretical applications connected to the Gauss map and to the Riemann $\zeta$-function illustrate our results.

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BibTeX RIS

Marc Kesseböhmer, Sabrina Kombrink. 2017-02-09. Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings. https://arxiv.org/abs/1702.02854

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