Escaping orbits are rare in the quasi-periodic Littlewood boundedness problem

We study the superlinear oscillator equation $\ddot{x}+ \lvert x \rvert^{\alpha-1}x = p(t)$ for $\alpha\geq 3$, where $p$ is a quasi-periodic forcing with no Diophantine condition on the frequencies and show that typically the set of initial values leading to solutions $x$ such that $\lim_{t\to\infty} (\lvert x(t) \rvert + \lvert \dot{x}(t) \rvert) = \infty$ has Lebesgue measure zero, provided the starting energy $\lvert x(t_0) \rvert + \lvert \dot{x}(t_0) \rvert$ is sufficiently large.