  We introduce modified energies that are suitable to get upper bounds on the
high Sobolev norms for solutions to the $1$D periodic NLS. Our strategy is
rather flexible and allows us to get a new and simpler proof of the bounds
obtained by Bourgain in the case of the quintic nonlinearity, as well as its
extension to the case of higher order nonlinearities. Our main ingredients are
a combination of integration by parts and classical dispersive estimates.
