Fourier non-uniqueness sets from totally real number fields

Let K be a totally real number field of degree n >= 2. The inverse different of K gives rise to a lattice in Rn. We prove that the space of Schwartz Fourier eigenfunctions on 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 root mS(n-1) for integers m >= 0 and, as m -> infinity, there are similar to 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.