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Analysis of PDEsCombinatoricsFunctional Analysismathscidoc: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.
@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.