Fourier non-uniqueness sets from totally real number fields

Abstract

Let $K$ be a totally real number field of degree $n\ge 2$. The inverse different of $K$ gives rise to a lattice in ${\mathbb{R}}^n$. We prove that the space of Schwartz Fourier eigenfunctions on ${\mathbb{R}}^n$ which vanish on the “component-wise square root” of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres $\sqrt{m}{S}^{n-1}$ for integers $m\ge 0$ and, as $m\to \infty$, there are $\sim c_K{m}^{n-1}$ many points on the $m$-th sphere for some explicit constant $c_K$, proportional to the square root of the discriminant of $K$. This contrasts a recent Fourier uniqueness result by Stoller (2021) Using a different construction involving the codifferent of $K$, we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each.

We also study a related question about existence of Fourier interpolation formulas with nodes “$\sqrt{\Lambda }$” for general lattices $\Lambda \subset {\mathbb{R}}^n$. Using results about lattices in Lie groups of higher rank we prove that if $n\ge 2$ and a certain group ${\Gamma }_{\Lambda }\le {\mathrm{PSL}}_2(\mathbb{R}{)}^n$ is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all $n\ge 5$ and all real $\lambda >2$, Fourier interpolation results for sequences of spheres $\sqrt{2m/\lambda }{S}^{n-1}$, where $m$ ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincaré type for Hecke groups of infinite covolume and is similar to the one in Stoller (2021).