Geometry Seminar / Graduate Lectures

There is a well-known hierarchy of regularity for continuous Z-periodic functions on Rⁿ,

C⁰⊃C¹⊃. . .⊃C^∞⊃analytic⊃trigonometric polynomials,

and the decay of Fourier coefficients precisely reflects this hierarchy. In particular, the Fourier transform ^f has finite support if and only if f is a trigonometric polynomial.

Periodic hypersurfaces in Rⁿ exhibit a similar hierarchy of regularity, yet no direct analogue of this Fourier characterization is known.

In this talk, I will present joint work with Mario Kummer in which we give a surprising sufficient Fourier criterion ensuring that a C^1+ε periodic hypersurface Σ⊂Rⁿ is the zero set of a trigonometric polynomial of the form

p(e^2πix₁, . . . , e^2πix_n),

where p is a Lee--Yang polynomial.

The criterion is formulated using a notion recently introduced by Yves Meyer: a periodic positive Radon measure m on Rⁿ is called a lighthouse measure if supp(m) has zero Lebesgue measure and supp(^m) is contained in a proper double cone.

Our proof produces explicit examples of such lighthouse measures and establishes an inverse theorem. It relies on recent breakthroughs concerning the existence and classification of one-dimensional Fourier quasicrystals. No field-specific background is assumed. This work builds on collaborations with Alex Cohen, Pavel Kurasov, and Cynthia Vinzant.