Sieving and expanders

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The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves (2007, background): “) (2) For any xed R>4=( ), jO(T;R)jT=logT: (1.5) Part (1) above tells us that the orbit is thin if and only if <1, and is the quintessence of our necessity to work in innite co-volume. In [BGS06], it is proved that if >1=2, then there always exists some spectral gap <. Gamburd [Gam02] sh”
Uniform expansion bounds for Cayley graphs of SL2(Fp) (2008, background): “In fact, our proof gives more than expansion (and this is important in applications [2]): if λ is an eigenvalue of A(G(SL2(Fp), Sp)), such that λ 6= ±2k, then |λ| ≤ κ < 2k where κ = κ(τ) is independent of p.”
Almost prime Pythagorean triples in thin orbits (2010, background): “In [4], Bourgain, Gamburd, and Sarnak introduced the a‰ne linear sieve, which extends some classical sieve methods to thin orbits of non-abelian group actions.”