Quasicrystals: rigidity and almost periodic functions

In our recent preprint [PK-X] we use modern results from arithmetic number theory to construct  positive crystalline measure and Fourier quasicrystals  using pairs of stable polynomials.

Such examples will definitely play an important role in classification of such exotic measures.Our major goal is on one side to extend our construction to get measures with more unusual properties and may be classify all positive crystalline measures, and on the other side to bring methods from Diophantine analysis to describe spectral properties of operators appearing in mathematical physics, in particular quantum graphs.

We expect to get explicit estimates on the length of arithmetic sequences that can be observed in the spectrum.

We shall also study different trace formulas and zeta-functions associated with metric graphs.The main applicant is planning to lead the project with certain partial goals delegated to a PhD student.

It is expected that methods from different areas of mathematics stretching from operator theory to number theory will be used.Planned research aims to fertilise spectral theory with new ideas coming from number theory most probably leading to surprising results.

Characterisation of crystalline measures is important for Fourier analysis, where crystalline measures in several variables are waiting their turn.