Small denominators and large numerators of quasiperiodic Schr\"odinger operators

We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schr\"odinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let $(H_{\lambda,\alpha,\theta}u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos2\pi(\theta+n\alpha)u(n)$ be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $\lambda, \alpha, \theta\in \mathbb{R}$. Let $$\beta(\alpha)=\limsup_{k\rightarrow \infty}-\frac{\ln ||k\alpha||_{\mathbb{R}/\mathbb{Z}}}{|k|}.$$ We prove that for any $\theta$ with $2\theta\in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $|\lambda|>e^{2\beta(\alpha)}$. This confirms a conjecture of Avila and Jitomirskaya [The Ten Martini Problem. Ann. of Math. (2) 170 (2009), no. 1, 303--342] and a particular case of a conjecture of Jitomirskaya [Almost everything about the almost Mathieu operator. II. XIth International Congress of Mathematical Physics (Paris, 1994), 373--382, Int. Press, Cambridge, MA, 1995].