An uncertainty principle for cyclic groups of prime order

Let $G$ be a finite abelian group, and let $f:G\to \text{C}$ be a complex function on $G$. The uncertainty principle asserts that the support $\supp(f) := {x \in G: f(x) \neq 0}$isrelatedtothesupportoftheFouriertransform$\hat f: G \to \C$bytheformula$$\mathrm{\mid }\text{supp}(f)\mathrm{\mid }\mathrm{\mid }\text{supp}(\widehat{f})\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }$$where$|X|$denotesthecardinalityof$X$\mathrm{.}Inthisnoteweshowthatwhen$G$isthecyclicgroup$\Z/p\Z$ofprimeorder$p$,thenwemayimprovethisto$$\mathrm{\mid }\text{supp}(f)\mathrm{\mid }+\mathrm{\mid }\text{supp}(\widehat{f})\mathrm{\mid }\ge p+1$$ and show that this is absolutely sharp. As one consequence, we see that a sparse polynomial in $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ consisting of $k+1$ monomials can have at most $k$ zeroes. Another consequence is a short proof of the well-known Cauchy-Davenport inequality.