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arXiv · 0912.5021

Generalization of Selberg's 3/16 Theorem and Affine Sieve

Jean Bourgain·Alex Gamburd·Peter Sarnak
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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.

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Jean Bourgain, Alex Gamburd, Peter Sarnak. 2009-12-26. Generalization of Selberg's 3/16 Theorem and Affine Sieve. https://arxiv.org/abs/0912.5021

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