Periodic Hypersurfaces and Lee-Yang Polynomials
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918466924052480 |
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| author | Alon, Lior Kummer, Mario |
| author_facet | Alon, Lior Kummer, Mario |
| contents | 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 $C^{1+ε}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Periodic Hypersurfaces and Lee-Yang Polynomials Alon, Lior Kummer, Mario Algebraic Geometry 52C23, 42B10, 42A75, 42A38, 14P05 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 $C^{1+ε}$ 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. |
| title | Periodic Hypersurfaces and Lee-Yang Polynomials |
| topic | Algebraic Geometry 52C23, 42B10, 42A75, 42A38, 14P05 |
| url | https://arxiv.org/abs/2507.16029 |