Generalization of Selberg's 3/16 Theorem and Affine Sieve
Number Theory
2009-12-31 v1 Spectral Theory
Abstract
A celebrated theorem of Selberg states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z). Consequently we obtain sharp upper bounds in the affine linear sieve, where in contrast to \cite{BGS} we use an archimedean norm to order the elements.
Keywords
Cite
@article{arxiv.0912.5021,
title = {Generalization of Selberg's 3/16 Theorem and Affine Sieve},
author = {Jean Bourgain and Alex Gamburd and Peter Sarnak},
journal= {arXiv preprint arXiv:0912.5021},
year = {2009}
}