  Let $G$ be a finite abelian group, and let $f: G \to \C$ be a complex
function on $G$. The uncertainty principle asserts that the support $\supp(f)
:= \{x \in G: f(x) \neq 0\}$ is related to the support of the Fourier transform
$\hat f: G \to \C$ by the formula $$ |\supp(f)| |\supp(\hat f)| \geq |G|$$
where $|X|$ denotes the cardinality of $X$. In this note we show that when $G$
is the cyclic group $\Z/p\Z$ of prime order $p$, then we may improve this to $$
|\supp(f)| + |\supp(\hat f)| \geq p+1$$ and show that this is absolutely sharp.
As one consequence, we see that a sparse polynomial in $\Z/p\Z$ consisting of
$k+1$ monomials can have at most $k$ zeroes. Another consequence is a short
proof of the well-known Cauchy-Davenport inequality.
