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We consider a class of linear time dependent Schr\"odinger equations andquasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon andSchr\"odinger equations on arbitrary flat tori. For the linear Schr\"odingerequation, we prove a $t^\epsilon$ $(\forall \epsilon >0)$ upper bound for thegrowth of the Sobolev norms as the time goes to infinity. For the nonlinearHamiltonian PDEs we construct families of time quasi-periodic solutions. Bothresults are based on "clusterization properties" of the eigenvalues of theLaplacian on a flat torus and on suitable "separation properties" of thesingular sites of Schr\"odinger and wave operators, which are integers, inspace-time Fourier lattice, close to a cone or a paraboloid. Thanks to theseproperties we are able to apply Delort abstract theorem [Del10] to control thespeed of growth of the Sobolev norms, and Berti-Corsi-Procesi abstractNash-Moser theorem [BCP15] to construct quasi-periodic solutions.

Long time dynamics of Schr\"odinger and wave equations on flat tori