Fourier uniqueness pairs of powers of integers

Abstract

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