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

By Jean Bourgain, Alex Gamburd, Peter Sarnak. 2009

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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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Citations in the corpus

An Improvement to Zaremba’s Conjecture (2013, background): “Bourgain-Gamburd-Sarnak [BGS11] later extended the work of Lalley’s to a congruence setting, and proved the following theorem.”
On the diameter of permutation groups (2011, background): “The paper [Hel08] initiated a period of intense activity [BG08a], [BG08b], [Din11], [BGS10], [Hel11], [GH], [Var12], [BGS11], [PS10], [BGT11], [GH10], [GV12] 1 on the diameter problem and the related problem of expansion properties of Cayley graphs.”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, unknown): “19 was previously obtained for n = 2, 3 and s ≤ 0 ([5], [33], [35], [34], [41]).”