Fourier non-uniqueness sets from totally real number fields

Let $K$ be a totally real number field of degree $n \geq 2$. The inverse different gives rise to lattice in $\mathbb{R}^n$. We prove that the space Schwartz Fourier eigenfunctions on $\mathbb{R}^n$ which vanish “component-wise square root” this lattice, is infinite dimensional. non-uniqueness set thus obtained discrete subset union all spheres $\sqrt{m}S^{n-1}$ for integers $m 0$ and, as \rightarrow \infty$, there are $\sim c\_{K} m^{n-1}$ many points $m$-th sphere some explicit constant $c\_{K}$, proportional root discriminant $K$. This contrasts recent uniqueness result by Stoller (2021) Using construction involving codifferent $K$, we an analogue subsets ellipsoids. In special cases, these sets also lie with more densely spaced radii, but fewer each. study related question about existence interpolation formulas nodes “$\sqrt{\Lambda}$” general lattices $\Lambda \subset \mathbb{R}^n$. results Lie groups higher rank if 2$ and certain group $\Gamma\_{\Lambda} \leq \operatorname{PSL}\_2(\mathbb{R})^n$ discrete, then such cannot exist. Motivated considerations, revisit case one radial variable prove, 5$ $\lambda > 2$, sequences $\sqrt{2 m/ \lambda}S^{n-1}$, where $m$ ranges over any fixed cofinite non-negative integers. proof relies series Poincaré type Hecke covolume similar (2021).