We study the self-adjoint Schr\"odinger operator on the axis
\[
H_v = -\frac{d^2}{dx^2} + v(x),\ \ \ -\infty < x < \infty,
\]
with an almost periodic real-valued potential $v(x)$.
Let $\Lambda$ be a dense subgroup of the group $(\R,+)$. Denote by
$AP_{\Lambda}(\mathbf{R})$ the Banach space of all real-valued almost periodic
functions on $\R$ whose all frequencies belong to $\Lambda$, with the supremum norm.
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\textbf{Theorem}
\ There exists a dense $G_{\delta}$ subset $X\subseteq AP_{\Lambda}(\mathbf{R})$,
such that for all $v\in X$ the operator $H_v$ has a nowhere dense spectrum.