  We investigate the zero set of a stationary Gaussian process on the real
line, and in particular give lower bounds for the variance of the number of
points on a large interval, in all generality. We prove that this point process
is never hyperuniform, i.e. the variance is at least linear, and give necessary
conditions to have linear variance, which are close to be sharp. We study the
class of symmetric Bernoulli convolutions and give an example where the zero
set is super rigid, weakly mixing, and not hyperuniform.
