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arXiv · 2507.16029

Periodic Hypersurfaces and Lee-Yang Polynomials

Lior Alon·Mario Kummer
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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.

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Lior Alon, Mario Kummer. 2025-07-21. Periodic Hypersurfaces and Lee-Yang Polynomials. https://arxiv.org/abs/2507.16029

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