Generalization of Selberg’s $$ \frac{3}{{16}} $$ theorem and affine sieve

By Jean Bourgain, Alex Gamburd, Peter Sarnak. 2011

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An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup L of SL(2,Z). The proof in the case that the Hausdorff of the limit set of L is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.

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

View Paper: https://www.semanticscholar.org/paper/4d33ab3c141df42dc97714a7865d0609eb57371a

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “This property always holds if δ > (n − 1)/2 for n = 2, 3 and if δ > n − 2 for n ≥ 4 via the recent works of Bourgain,Gamburd, Sarnak ([6], [5]) and Salehi-Golsefidy and Varju [58] together with the classification of the unitary dual of G (see Theorem 8.”