Anderson localization for the completely resonant phases

For the almost Mathieu operator (Hλ,α,θu)(n)=u(n+1)+u(n−1)+2λcos⁡2π(θ+nα)u(n)(Hλ,α,θu)(n)=u(n+1)+u(n−1)+2λcos⁡2π(θ+nα)u(n), Avila and Jitomirskaya conjecture that for every phase θ∈R≜{θ∈R|2θ+αZ∈Z}θ∈R≜{θ∈R|2θ+αZ∈Z}, Hλ,α,θHλ,α,θ satisfies Anderson localization if |λ|>e2β|λ|>e2β. In the present paper, we verify the conjecture for |λ|>e7β|λ|>e7β.