Let
$\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$
and
$\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=\limsup _{n\rightarrow \infty }(\ln q_{n+1})/q_{n}<\infty$
, where
$p_{n}/q_{n}$
is the continued fraction approximation to
$\unicode[STIX]{x1D6FC}$
. Let
$(H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2\unicode[STIX]{x1D706}\cos 2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}+n\unicode[STIX]{x1D6FC})u(n)$
be the almost Mathieu operator on
$\ell ^{2}(\mathbb{Z})$
, where
$\unicode[STIX]{x1D706},\unicode[STIX]{x1D703}\in \mathbb{R}$
. Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170(1) (2009), 303–342] conjectured that, for
$2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$
,
$H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$
satisfies Anderson localization if
$|\unicode[STIX]{x1D706}|>e^{2\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$
. In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for
$2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$
,
$H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$
satisfies Anderson localization if
$|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$
.