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On the local-global principle for integral Apollonian 3-circle packings

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Veröffentlicht/Copyright: 20. August 2015

Abstract

In this paper we study the integral properties of Apollonian 3-circle packings, which are variants of the standard Apollonian circle packings. Specifically, we study the reduction theory, formulate a local-global conjecture, and prove a density one version of this conjecture. Along the way, we prove a uniform spectral gap for the congruence towers of the symmetry group.

Award Identifier / Grant number: DMS-1209373

Award Identifier / Grant number: DMS-1064214

Award Identifier / Grant number: DMS-1001252

Award Identifier / Grant number: DMS-1254788

Funding statement: The author acknowledges support for this work from Prof. Kontorovich’s NSF grants DMS-1209373, DMS-1064214, DMS-1001252, and NSF CAREER grant DMS-1254788.

Acknowledgements

This paper is essentially the content of the author’s Ph.D. thesis when he was a graduate student at Stony Brook. The author is greatly indebted to his PhD advisor Prof. Alex Kontorovich for introducing this beautiful subject to the author and for numerous enlightening discussions. The author also thanks the referee for her/his numerous corrections and helpful suggestions. In writing up this paper, the author utilizes the codes provided by Prof. Kontorovich for several pictures.

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Received: 2014-6-8
Revised: 2015-4-20
Published Online: 2015-8-20
Published in Print: 2018-4-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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