Pavel Kurasov: On Higher Dimensional Crystalline Measures and Fourier Quasicrystals [moved]

Abstract:

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 \[|\mu| : = \sum_{\lambda \in \Lambda} |a_\lambda| \delta_\lambda \quad \text{and} \quad |\hat{\mu}| := \sum_{s \in S} |b_s| \delta_s,\]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 (Inventiones Mathematicae, (2024) 239:321–376).