Let $f : \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let $\hat{f} : \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ be its DFT given by $\hat{f}(m) = \sum_{j \in \mathbb{Z}/N\mathbb{Z}} f(j)e_N(jm)$, where $e_N(x) := e^{2\pi i x/N}$. If $\hat{f}(m) = 0$ for every $m$ in some interval $[M+1, M+K]$ with $K \geq \sqrt{N}$, is there anything interesting we can say about $f$ and its support?
I am aware of the relation $\#\text{supp}(f) \#\text{supp}(\hat{f}) \geq N$, sometimes called the Heisenberg uncertainty inequality. But I wonder if the more specific information, that $\hat{f}$ vanishes on a large interval, implies anything interesting about $f$? Thanks.