  We investigate the asymptotic variance of Gaussian nodal excursions in the
Euclidean space, focusing on the case where the spectral measure has
incommensurable atoms. This study requires to establish fine recurrence
properties in 0 for the associated irrational random walk on the torus. We show
in particular that the recurrence magnitude depends strongly on the diophantine
properties of the atoms, and the same goes for the variance asymptotics of
nodal excursions. More specifically, if the spectral measures contains atoms
which ratios are well approximable by rationals, the variance is likely to have
large fluctuations as the observation window grows, whereas the variance is
bounded by the (d -- 1)-dimensional measure of the window boundary if these
ratio are badly approximable. We also show that, given any reasonable function,
there are uncountably many sets of parameters for which the variance is
asymptotically equivalent to this function.
