Long-time existence for semi-linear Klein–Gordon equations on tori

We use normal forms for Sobolev energy to prove that small smooth solutions of semi-linear Klein–Gordon equations on the torus exist over a larger interval than the one given by local existence theory, for almost every value mass. The gain on the length of the lifespan does not depend on the dimension. The result relies on the fact that the difference of square of two successive distinct eigenvalues of on Td can be bounded from below by a constant.