Periodic Hypersurfaces and Lee-Yang Polynomials

TL;DR

This paper characterizes periodic hypersurfaces supporting lighthouse measures as essentially being zero sets of Lee-Yang polynomials, linking Fourier analysis, algebraic geometry, and recent quasicrystal classification.

Contribution

It proves a rigidity theorem showing such hypersurfaces must originate from Lee-Yang polynomials under mild conditions.

Findings

Periodic hypersurfaces supporting lighthouse measures are characterized as Lee-Yang polynomial zero sets.

The proof leverages recent classification of Fourier quasicrystals.

Provides a geometric interpretation connecting Fourier analysis and algebraic geometry.

Abstract

We study periodic measures on $\mathbb{R}^n$ whose Fourier transform is confined to a proper double cone, in the sense of Meyer's notion of lighthouse measures. Lee--Yang polynomials provide a natural family of examples: it follows from the work of Kurasov and Sarnak that the torus zero sets of such polynomials are hypersurfaces supporting directional lighthouse measures. We prove a rigidity theorem showing that, under mild assumptions, this is essentially the only possibility. Any periodic $C^{1+\epsilon}$ hypersurface supporting a directional lighthouse measure must arise as the torus zero set of an essentially Lee--Yang polynomial. The proof is based on the recent classification of one-dimensional Fourier quasicrystals and provides a geometric interpretation of this theory.