Periodic Hypersurfaces and Lee-Yang Polynomials

We provide a sufficient Fourier-criterion for a C1+ε-hypersurface in the n-dimensional torus Tⁿ to be the zero set of a (Lee--Yang) polynomial. The criterion can be stated using a notion recently introduced by Yves Meyer: a positive Radon measure m on Tⁿ is a lighthouse measure if supp(m) has zero Lebesgue measure and supp(m) is contained in a proper double cone. Our proof relies on the classification of one-dimensional Fourier quasicrystals and adds a geometric interpretation.