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We consider the cubic nonlinear Schr\"odinger equation on product manifolds$\mathbb{R}\times \mathbb{T}$. In this paper, we obtain polynomial bounds onthe growth in time of high Sobolev norms of the solutions. The main ingredientof the proof is to establish an iteration bound, which is based on the ideaused by Bourgain in \cite{B1}.

On the growth of high Sobolev norms of the cubic nonlinear Schr\"odinger equation on $\mathbb{R}\times \mathbb{T}$