Fourier non-uniqueness sets from totally real number fields

TL;DR

This paper investigates the structure of Fourier eigenfunctions that vanish on specific lattice-derived sets in Euclidean space, revealing infinite-dimensional spaces and contrasting with recent uniqueness results, with implications for Fourier interpolation and lattice theory.

Contribution

It introduces new examples of Fourier non-uniqueness sets derived from totally real number fields and explores their properties, contrasting with existing Fourier uniqueness results.

Findings

Infinite-dimensional space of Fourier eigenfunctions vanishing on lattice-derived sets.

Asymptotic count of points on spheres related to the discriminant of the number field.

Non-existence of certain Fourier interpolation formulas under specific group discreteness conditions.

Abstract

Let $K$ be a totally real number field of degree $n \geq 2$. 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 $m \geq 0$ and, as $m \rightarrow \infty$, 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. Using a different construction involving the codifferent of $K$, we prove an analogue of our results 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 $n \geq 2$ and if a certain group $\Gamma_{\Lambda} \leq \operatorname{PSL}_2(\mathbb{R})^n$ 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 $n \geq 5$ and all real $\lambda > 2$, Fourier interpolation results for sequences of spheres $\sqrt{2 m/ \lambda}S^{n-1}$, where $m$ ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincar\'e type for Hecke groups of infinite covolume, similarly to the construction previously used by Stoller.