Sector Estimates for Hyperbolic Isometries

By Jean Bourgain, Alex Kontorovich, Peter Sarnak. 2010

62 citations recorded by the source corpus.

We prove various orbital counting statements for Fuchsian groups of the second kind. These are of independent interest, and also are used in the companion paper [BK] to produce primes in the Affine Linear Sieve.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02c0d87612abe0a97bec08cd00f4c8d857043973

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Citations in the corpus

On representations of integers in thin subgroups of SL(2,Z) (2010, methodology): “)j<N˙ and jyj<N1=2 ˙j(c;d)j. Then X 2 k k<N1=2 ˙ 1 ˆ j(c;d) yj< N1=2 K ˙ 1 ˆ (c;d) y(modq) ˙ ˝ N(1 2˙) K1+q2 + N(1=2 ˙)(6 7 2+ 5 21 ); as N!1. Proof. This follows from Theorem 1.15 in [BKS09].INTEGERS IN THIN SUBGROUPS OF SL 2(Z) 9 3. Setup of the Exponential Sum and Major/Minor Arcs 3.1”
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits (2012, methodology): “; >> >> >> >: innite volume spectral and representation theory a la x3.1.6, specically Vinogradov’s thesis [Vin12], if X= A, similar techniques developed by Bourgain-K.-Sarnak [BKS10], if X= P, the thermodynamic formalism, analytically continuing certain Ruelle transfer operator”
On the local-global conjecture for integral Apollonian gaskets (2012, result): “rson-Sullivan measure (cf. (4.5)), and < (4.16) depends only on the spectral gap for . The implied constant does not depend on ;#TAB#;or I. This generalizes from SL(2;R) to SL(2;C) the main result of [BKS10], which is itself a generalization (with weaker exponents) to our innite volume setting of”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “14 was earlier shown in [9] and [64] for n = 2, 3 respectively.”