In this paper, the author proves the existence of infinitely many time-periodic solutions of nonlinear Schrödinger equations by means of pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov-Floer compactness theorem to infinite dimensions, the author shows how to solve the arising small divisor problem by combining elliptic methods with results from the theory of diophantine approximations.