Fourier uniqueness pairs of powers of integers (Q2098199)

Summary: 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 \(n \geq 0\) must vanish identically, complementing a series of recent results involving uncertainty principles, such as the pointwise interpolation formulas by \textit{D. Radchenko} and \textit{M. Viazovska} [Publ. Math., Inst. Hautes Étud. Sci. 129, 51--81 (2019; Zbl 1455.11075)] and the Meyer-Guinnand construction of self-dual crystaline measures (see [\textit{Y. Meyer}, Rev. Mat. Iberoam. 33, No. 3, 1025--1036 (2017; Zbl 1384.42009)]).