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

Kontorovich, Alex

doi:10.1090/s0273-0979-2013-01402-2

Cited by 54 publications

(47 citation statements)

References 67 publications

(54 reference statements)

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“…The reader can also see the survey of Kontorovich [14] that situates Zaremba's conjecture amongst other problems in the 'thin (semi)groups' setting.…”

Section: Conjecture 3 (Zaremba)mentioning

confidence: 99%

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Magee

Winter

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Citation for published item:wgeeD wihel nd yhD ree nd interD hle @PHIUA 9niform ongruene ounting for hottky semigroups in vP@AF9D tournl f¤ ur die reine und ngewndte wthemtikF a grelles journlF F Further information on publisher's website:The nal publication is available at www.degruyter.com With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let Γ be a Schottky semigroup in SL2(Z), and for q ∈ N, let Γ(q) := {γ ∈ Γ : γ = e (mod q)} be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls BR in M2(R) of radius R: for all positive integer q with no small prime factors,as R → ∞ for some cΓ > 0, C > 0, > 0 which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of SL2(Z), which arises in the study of Zaremba's conjecture on continued fractions.

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“…The reader can also see the survey of Kontorovich [14] that situates Zaremba's conjecture amongst other problems in the 'thin (semi)groups' setting.…”

Section: Conjecture 3 (Zaremba)mentioning

confidence: 99%

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Magee

Winter

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…It turns out that one can do much more using recent progress on "local-global"problems in "thin orbits" (see, e.g., the discussion in [Kon13]); namely, one can produce not just prime divisors but actual primes in the entries of Γ, and moreover give explicit estimates for their exponential growth rates (which are far superior compared to those which would come from an Affine Sieve analysis). Our main result is the following Theorem A.1.…”

Section: James Conjecturementioning

confidence: 99%

Schubert calculus and torsion explosion

Williamson

2016

J. Amer. Math. Soc.

104

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Abstract. We observe that certain numbers occurring in Schubert calculus for SLn also occur as entries in intersection forms controlling decompositions of Soergel bimodules in higher rank. These numbers grow exponentially. This observation gives many counter-examples to the expected bounds in Lusztig's conjecture on the characters of simple rational modules for SLn over fields of positive characteristic. Our examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.

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“…The Descartes Circle Theorem has been popular lately because it underpins the geometry and arithmetic of Apollonian packings, a subject of great current interest; see, e.g., surveys [7,8,9,13]. In this article we revisit this classic result, along with another old theorem on circle packing, the Steiner porism, and relate these topics to spherical designs.…”

Section: Introductionmentioning

confidence: 97%