Fourier uniqueness pairs of powers of integers

Abstract

We prove, under certain conditions on $(\alpha ,\beta )$, that each Schwartz function $f$ such that $f(\pm n^{\alpha })=\hat{f}(\pm n^{\beta })=0$ for all $n\ge 0$ 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.