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

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.

Cited by in corpus (5)

• Expander graphs, gonality and variation of Galois representations
• A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions
• Expander Graphs in Pure and Applied Mathematics
• Sieve methods in group theory I: Powers in Linear groups
• Growth in groups: ideas and perspectives