  We study uncertainty principles for function classes on the torus. The
classes are defined in terms of spectral subspaces of the energy or the
momentum, respectively. In our main theorems, the support of the Fourier
transform of the considered functions is allowed to be supported in a (finite
number of) parallelepipeds. The estimates we obtain do not depend on the size
of the torus and the position of the parallelepipeds, but only on their size
and number, and the density and scale of the observability set. Our results are
on the one hand closely related to unique continuation for linear combinations
of eigenfunctions (aka spectral inequalities) which can be obtained by Carleman
estimates, on the other hand to observability estimates for the time-dependent
Schroedinger and for the heat equation, and finally to the Logvinenko & Sereda
theorem. In fact, they are based on the methods developed by Kovrijkine to
refine and generalize the results of Logvinenko & Sereda and Kacnel'son.
Furthermore, relying on completely different techniques associated with the
time-dependent Schroedinger equation, we prove a companion theorem where the
energy of the considered functions is allowed to be in a spectral subspace of a
Schroedinger operator.
