  Generalized uncertainty principles are able to serve as useful descriptions
of some of the phenomenology of quantum gravity effects, providing an intuitive
grasp on non-trivial space-time structures such as a fundamental discreteness
of space, a universal bandlimit or an irreducible extendedness of elementary
particles. In this article, uncertainty relations are derived by a moment
expansion of states for quantum systems with a discrete coordinate, and
correspondingly a periodic momentum. Corrections to standard uncertainty
relations are found, with some similarities but also key differences to what is
often assumed in this context. The relations provided can be applied to
discrete models of matter or space-time, including loop quantum cosmology.
