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We study the Cauchy problem for the $1$-d periodic fractional Schr\"odingerequation with cubic nonlinearity. In particular we prove local well-posednessin Sobolev spaces, for solutions evolving from rough initial data. In additionwe show the existence of global-in-time infinite energy solutions. Our toolsinclude a new Strichartz estimate on the torus along with ideas that Bourgaindeveloped in studying the periodic cubic NLS.

Existence and Uniqueness theory for the fractional Schr\"odinger equation on the torus