Fourier uniqueness pairs of powers of integers

We prove, under certain conditions on $(\alpha,\beta)$, that each Schwartz function $f$ such that $f(\pm n^{\alpha}) = \hat{f}(\pm n^{\beta}) = 0, \forall n \ge 0$ must vanish identically, complementing a series of recent results involving uncertainty principles, such as the pointwise interpolation formulas by Radchenko and Viazovska and the Meyer-Guinnand construction of self-dual crystaline measures.