  We establish an uncertainty principle for functions $f: \mathbb{Z}/p
\rightarrow \mathbb{F}_q$ with constant support (where $p \mid q-1$). In
particular, we show that for any constant $S > 0$, functions $f: \mathbb{Z}/p
\rightarrow \mathbb{F}_q$ for which $|\text{supp}\; {f}| = S$ must satisfy
$|\text{supp}\; \hat{f}| = (1 - o(1))p$. The proof relies on an application of
Szemeredi's theorem; the celebrated improvements by Gowers translate into
slightly stronger statements permitting conclusions for functions possessing
slowly growing support as a function of $p$.
