Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

Fourier Quasicrystals (FQ) are defined as crystalline measures

μ=∑λ∈Λaλδλ,μ̂ =∑s∈Sbsδs, so that not only μ (and hence μ̂ ) are tempered distributions, but also |μ|:=∑λ∈Λ|aλ|δλand|μ̂ |:=∑s∈S|bs|δs,

are tempered.

One-dimensional FQs with positive integer weights (that is aλ∈ℕ) can be described using stable Lee-Yang polynomials, as was proven in a joint work with Peter Sarnak. Multidimensional Fourier quasicrystals are discussed in the current talk. It is shown that a rather general family of FQs in ℝd

with positive integer weights can be constructed using co-dimension d Lee-Yang varieties in ℂn,n greater than d.

These complex algebraic varieties are symmetric and avoid certain regions in ℂn, thus generalising zero sets of Lee-Yang polynomials.

It is shown that such FQs can be supported by Delaunay almost periodic sets and are genuinely multidimensional in the sense that their restriction to any one-dimensional subspace is not given by a one-dimensional FQ. Connections to alternative recent approaches by Yves Meyer, Lawton-Tsikh and de Courcy-Ireland-K. are clarified. Is it possible that our construction gives all multidimensional FQs with positive integer masses?

This is joint work with L. Alon, M. Kummer, and C. Vinzant (arXiv:2407.11184).