Generalization of Selberg’s $ \frac{3}{{16}} $ theorem and affine sieve

Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $\operatorname {SL}_2(\mathbb {Z})$

“…The bounds for L ab in this region follow immediately from the complex Ruelle-Perron-Frobenius theorem and a compactness argument. Since we require bounds on M ab,q uniformly in q, however, this compactness argument is not available to us; instead we follow the approach and use the expansion machinery of Bourgain-Gamburd-Sarnak [8].…”

“…The key ingredient of the proof of Theorem 4.1 is the expander technology, introduced in this context by Bourgain, Gamburd, and Sarnak [8], from which we draw heavily throughout this section. The idea of the expansion machinery is that random walks on the Cayley graphs of SL 2 (q) have good spectral properties.…”

“…If Γ < SL 2 (Z) is finitely generated with δ > 1 2 , then a version of Theorem 1.1 is known by [8] and [12] with a different interpretation of the main term (also see [22], [43], [27]). Therefore the main contribution of Theorem 1.1 lies in the groups Γ with δ ≤ 1 2 ; such groups are known to be convex cocompact.…”

“…Then φ o (x) := |ν x | is an eigenfunction of ∆ in C ∞ (Γ(q)\H 2 ) with eigenvalue δ(1 − δ) [31], and φ o ∈ L 2 (Γ(q)\H 2 ) if and only if δ Γ > 1/2. If we identity H 2 = SL 2 (R)/ SO (2), and ψ ∈ C c (Γ(q)\G) is SO (2) [8] established a uniform spectral gap for the smallest two Laplace eigenvalues on L 2 (Γ(q)\H 2 ) for all square-free q ∈ N with no small prime divisors; for some ǫ > 0, there are no eigenvalues between δ(1 − δ), which is known to be the smallest one, and δ(1 − δ) + ǫ. When δ ≤ 1 2 , the L 2 -spectrum of ∆ is known to be purely continuous [23], and the relevant spectral quantities are the resonances.…”

“…Naud proved that for some ǫ(q) > 0, the half-plane ℜs > δ − ǫ(q) is a resonance-fee region except at s = δ [28]. Bourgain, Gamburd and Sarnak showed that for some ǫ > 0, {ℜs > δ − ǫ · min{1, 1/(log(1 + |ℑs|)}} is a resonance free region except for s = δ, for all square-free q with no small prime divisors [8]. We will deduce a uniform resonance-free half plane from Theorem 1.1 (see Section 6): Theorem 1.3.…”