Speaker: Lior Alon (MIT)

Title: Periodic Hypersurfaces, Lighthouse Measures, and Lee–Yang Polynomials

Abstract: There is a hierarchy of regularity for continuous ℤ^𝑛 -periodic
functions in ℝ^𝑛 , 𝐶⁰ ⊃ 𝐶¹ ⊃ ⋯ ⊃ 𝐶^∞ ⊃ analytic ⊃ trigonomet-
ric polynomial, and the decay of the Fourier coefficients pre-
cisely reflects this regularity. In particular, the support supp(f̂)
is finite if and only if 𝑓 is a trigonometric polynomial. Periodic
hypersurfaces in ℝ^𝑛 exhibit a similar regularity hierarchy, but
there is no analogous Fourier description.

In this talk, I will present a joint work with Mario Kummer in
which we provide a sufficient Fourier-criterion for a 𝐶^1+𝜖 peri-
odic hypersurface Σ ⊂ ℝ^𝑛 to be the zero set of a trigonomet-
ric polynomial of the form 𝑝(𝑒^{2𝜋𝑖𝑥₁}, … , 𝑒^{2𝜋𝑖𝑥_𝑛} ) with 𝑝 Lee–Yang
polynomial.

The criterion can be stated using a recent notion introduced by
Yves Meyer: a periodic and positive Radon measure 𝑚 on ℝ^𝑛
is a lighthouse measure if supp(𝑚) has zero Lebesgue measure
and supp(m̂) is contained in a proper double cone.

Our proof relies on the classification of one-dimensional Fourier
quasicrystals. No field specific background is assumed. This
work is based on collaborations with Alex Cohen, Pavel Kurasov,

and Cynthia Vinzant.