Motivated by polynomial identity testing with exponentials (Li and Wu, ITCS'26), we study uncertainty principles for the number-theoretic transform (NTT). We show that the NTT satisfies strong sparsity tradeoffs: For every fixed prime $q$ and for all but finitely many primes $p \equiv 1 \pmod q$ every nonzero $f\in \mathbb F_p^{\mathbb Z_q}$ and its number-theoretic transform $\hat f$ satisfy \[ |\mathrm{Supp}(f)| + |\mathrm{Supp}(\hat f)| \ge q+1. \] Thus, a $k$-sparse function has transform support at least $q-k+1$. As our main technical contribution, we prove a probabilistic version of the above uncertainty principle, averaged over primes $p$, in the regime $p=q^{O(1)}$. As an application, we obtain a black-box identity test for $k$-sparse exponential polynomials of degree at most $d$ with vanishing soundness error, for $q$ moderately larger than $k$.

翻译：受指数多项式恒等测试（Li and Wu, ITCS'26）的启发，我们研究了数论变换（NTT）的不确定性原理。我们证明NTT满足强稀疏性权衡：对于每个固定素数$q$以及所有除有限多个外满足$p \equiv 1 \pmod q$的素数$p$，每个非零函数$f\in \mathbb F_p^{\mathbb Z_q}$及其数论变换$\hat f$满足\[ |\mathrm{Supp}(f)| + |\mathrm{Supp}(\hat f)| \ge q+1. \] 因此，一个$k$-稀疏函数的变换支撑集至少为$q-k+1$。作为主要技术贡献，我们在$p=q^{O(1)}$的范围内，证明了上述不确定性原理在素数$p$上取平均的概率版本。作为应用，我们获得了一个关于度数至多为$d$的$k$-稀疏指数多项式的黑盒恒等测试，该测试在$q$略大于$k$的情况下具有零可靠度误差。