On representations of integers in thin subgroups of SL(2,Z)

By Jean Bourgain, Alex Kontorovich. 2010

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Let Gamma , for gamma in Gamma. Assume that the limit set of Gamma has Hausdorff dimension delta>0.99995, that is, Gamma is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd's 5/6-th spectral gap in infinite-volume, we show that S contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set E(N) of integers |n| =1) but fail to be globally represented, n is not in S, has a power savings, $|E(N)| 0.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5177de5e436d8855b2ab96b328835574c975518

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits (2012, unknown): “Theorem P (Bourgain and Kontorovich, 2010 [BK10]).”
Sector Estimates for Hyperbolic Isometries (2010, methodology): “ X γ∈Γ k<T f(γ)+Eq′ +O (1+S∞,Tf) 6/7T6 7 ·2δ+1 7 ·2Θ . Here Eq′ ≪ T2δ−α0, with α 0 >0, and all implied constants are independent of q′′ and γ 0. The application in the companion paper [BK09] requires the following two estimates, which we derive as consequences of the above. Theorem 1.14. ”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, methodology): “We also formulate our counting statements for bisectors in the HAK decomposition, motivated by recent applications in [7] and [8].”