We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma}_{T}$, $\mathbb{P}\left({\Sigma}_{T}\right)>0$ such that any data ${\phi}^{\omega}\left(x\right)\in {H}^{\gamma}\left({\mathbb{T}}^{3}\right),\gamma <1,\omega \in {\Sigma}_{T}$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-{e}^{it\Delta}{\phi}^{\omega}\in C([0,T];{H}^{s}\left({\mathbb{T}}^{3}\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space ${H}^{1}\left({\mathbb{T}}^{3}\right)$, that is in the supercritical scaling regime.