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

Jean Bourgain School of Mathematics, Institute for Advanced Study Alex Gamburd Department of Mathematics, University of California at Santa Cruz Peter Sarnak School of Mathematics, Institute for Advanced Study

Analysis of PDEs Combinatorics Functional Analysis mathscidoc:1701.03005

Acta Mathematica, 207, (2), 255-290, 2009.12
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.
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@inproceedings{jean2009generalization,
  title={Generalization of Selberg’s $$ \frac{3}{{16}} $$ theorem and affine sieve},
  author={Jean Bourgain, Alex Gamburd, and Peter Sarnak},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357420511746},
  booktitle={Acta Mathematica},
  volume={207},
  number={2},
  pages={255-290},
  year={2009},
}
Jean Bourgain, Alex Gamburd, and Peter Sarnak. Generalization of Selberg’s $$ \frac{3}{{16}} $$ theorem and affine sieve. 2009. Vol. 207. In Acta Mathematica. pp.255-290. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357420511746.
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