Anderson Localization for the Almost Mathieu Operator in Exponential Regime

For the almost Mathieu operator $(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda \cos2\pi(\theta+n\alpha)u_n$, Avila and Jitomirskaya guess that for a.e. $\theta$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $|\lambda| > e^{ \beta}$, and they establish this for $|\lambda| > e^{\frac{16}{9} \beta}$. In the present paper, we extend their result to regime $|\lambda| > e^{\frac{3}{2} \beta}$.