Strong approximation in the Apollonian group

Fuchs, Elena

doi:10.1016/j.jnt.2011.05.010

Cited by 10 publications

(27 citation statements)

References 12 publications

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“…They observe empirically that congruence obstructions for any integral gasket seem to be to the modulus 24, and this is completely clarified (as we explain below) by Fuchs [Fuc11] in her thesis. Further convincing numerical evidence toward the conjecture is given in Fuch and Sanden [FS11].…”

Section: Conjecture a Every Sufficiently Large Admissible Number Is mentioning

confidence: 81%

“…Then changing variables fromQ back to the Descartes form Q by a conjugation, one gets the desired map ρ : SL 2 (C) → SO Q (R). It is straightforward then to compute the pullback of Γ ∩ SO Q under ρ (see [GLM05,Fuc11]), the answer being the following Lemma 3.32. There is 4 a homomorphism ρ : SL 2 (C) → SO Q (R) so that the group…”

Section: Theorem a (Bourgain And Kontorovich 2012 [Bk12]) Almost Ementioning

confidence: 99%

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From Apollonius to Zaremba: Local-global phenomena in thin orbits

Kontorovich

2013

Bull. Amer. Math. Soc.

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Abstract. We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving "thin" orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba's Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.

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Section: Conjecture a Every Sufficiently Large Admissible Number Is mentioning

confidence: 81%

Section: Theorem a (Bourgain And Kontorovich 2012 [Bk12]) Almost Ementioning

confidence: 99%

From Apollonius to Zaremba: Local-global phenomena in thin orbits

Kontorovich

2013

Bull. Amer. Math. Soc.

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“…Indeed, this remarkable integrality feature gives rise to several natural questions about integer ACPs; Graham et al make some progress towards answering them in [27] and pose striking conjectures, many of which are now theorems or at least better understood (see [7], [8], [9], [13], [19], [20], [21], [22], [32], [46], etc.) In this article we will survey how all these questions are handled and give an overview of what is currently known.…”

Section: Theorem 12 (Descartes 1643) Let a B C And D Denote Thementioning

confidence: 99%

“…One such result is the following. In Section 2 we review the results in [21] which extend Theorem 1.5 and give a complete answer to Question 1, namely it is shown that the only congruence obstructions for any primitive integer ACP are modulo 24, and that the number 30 in Theorem 1.5 above can be improved to 6. The basic idea is to use the representation of the packing as an orbit of the Apollonian group A and analyze the mod d structure of A.…”

Section: Questionmentioning

confidence: 99%

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Counting problems in Apollonian packings

Fuchs

2013

Bull. Amer. Math. Soc.

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Abstract. An Apollonian circle packing is a classical construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature, making the packings of interest from a number theoretic point of view. Many of the natural arithmetic problems have required new and sophisticated tools to solve them. The reason for this difficulty is that the study of Apollonian packings reduces to the study of a subgroup of GL 4 (Z) that is thin in a sense that we describe in this article, and arithmetic problems involving thin groups have only recently become approachable in broad generality. In this article, we report on what is currently known about Apollonian packings in which all circles have integer curvature and how these results are obtained. This survey is also meant to illustrate how to treat arithmetic problems related to other thin groups.

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A Geometric Study of Circle Packings and Ideal Class Groups

Martin

2024

Discrete Comput Geom

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Strong approximation in the Apollonian group

Cited by 10 publications

References 12 publications

From Apollonius to Zaremba: Local-global phenomena in thin orbits

From Apollonius to Zaremba: Local-global phenomena in thin orbits

Counting problems in Apollonian packings

A Geometric Study of Circle Packings and Ideal Class Groups

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