Local and global properties of approximate lattices

Research Project, 2020 – 2023

Approximate lattice were first introduced by Tobias Hartnick and myself in our paper "Approximate lattices" (Duke 2018). They can be viewed as a merger of approximate groups (in the sense of Tao) and lattices (discrete subgroups of locally compact groups with finite covolume). In Euclidean spaces they are more commonly known as quasicrystals. This project aims to continue the study of these objects, with a special emphasis on their dynamical and arithmetic properties. In Euclidean spaces (and more recently, nilmanifolds) it is known that approximate lattices are contained in cut-and-project sets, which are certain projections of higher dimensional lattices. This is a very useful global feature, which often allows us to reduce questions about them to lattice questions, where much more is known. Half of this project is concerned with various conjectural generalizations of this observation, mostly in connection with arithmeticity problems of approximate lattices in higher rank Lie groups.The second part of the project deals with spectral (local) properties of approximate lattices, in particular in relation with diffraction theory. Here we propose to study spherical and discrete series diffraction of approximate lattices in Heisenberg groups and symmetric spaces; the aim is to better understand the diffraction measures in this setting and how the corresponding irreducible representations embed into various function spaces of the hull of the approximate lattice.