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We establish an uncertainty principle for functions $f: \mathbb{Z}/p\rightarrow \mathbb{F}_q$ with constant support (where $p \mid q-1$). Inparticular, 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 ofSzemeredi's theorem; the celebrated improvements by Gowers translate intoslightly stronger statements permitting conclusions for functions possessingslowly growing support as a function of $p$.

Small-Support Uncertainty Principles on $\mathbb{Z}/p$ over Finite Fields