  We consider the long-term dynamics of nonlinear dispersive waves in a finite
periodic domain. The purpose of the work is to show that the statistical
properties of the wave field rely critically on the structure of the discrete
resonant manifold (DRM). To demonstrate this, we simulate the two-dimensional
MMT equation on rational and irrational tori, resulting in remarkably different
power-law spectra and energy cascades at low nonlinearity levels. The
difference is explained in terms of different structures of the DRM, which
makes use of recent number theory results.
