Analysis and Mathematical Physics

Fourier Quasicrystals (FQ) are defined as crystalline measures $$ \mu = \sum_{\lambda \in \Lambda} a_\lambda \delta_\lambda, \quad \hat{\mu} = \sum_{s \in S} b_s \delta_s, $$ so that not only $ \mu $ (and hence $ \hat{\mu} $) are tempered distributions, but also $$ |\mu| : = \sum_{\lambda \in \Lambda} |a_\lambda| \delta_\lambda \quad \mbox{and} \quad |\hat{\mu}| := \sum_{s \in S} |b_s| \delta_s, $$ are tempered. 

One-dimensional FQs with positive integer weights (that is $a_\lambda \in \mathbb 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 $ \mathbb R^d $ with positive integer weights can be constructed using co-dimension $d$ Lee-Yang varieties in $ \mathbb C^n, \, n > d.$ These complex algebraic varieties are symmetric and avoid certain regions in $ \mathbb C^n $, 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).