Small-Support Uncertainty Principles on Z / p over Finite Fields

where f̂ : Z/p → C denotes the Fourier transform of f , and supp g denotes the support of the function g. It is easy to check that inequality (1) is tight for any characteristic function 1H of a subgroup H of G. More specifically, (1) is tight precisely when the function f is the scaled characteristic function of a coset of a subgroup. This suggests that the simple groups Z/p—having no nontrivial subgroups—may play a special role in the theory and, in particular, may enjoy a stronger uncertainty principle. Indeed, a recent article of Tao [3] establishes a remarkable strengthening over these groups: any nonzero f : Z/p → C satisfies