Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals

Faustin Adiceam,Alan Haynes,David Damanik,Uwe Grimm,Franz Gähler,Lorenzo Sadun,Barak Weiss,Antoine Julien,Andrés Navas

doi:10.1007/s40598-016-0046-6

Abstract

This list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems below reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field.

Highlights

Consult Baake and Grimm (2013) or Sadun (2008), as well as the references provided with the relevant problem
A Schrödinger equation associated with a Schrödinger operator H can be used to model how well quantum wave packets travel in a quasicrystal—see Damanik and Tcheremchantsev (2010) for details
For cutand-project sets and tilings, this is the case by construction, but for inflation tilings the situation is not as clear

Summary

Problems

A Schrödinger equation associated with a Schrödinger operator H can be used to model how well quantum wave packets travel in a quasicrystal—see Damanik and Tcheremchantsev (2010) for details. For cutand-project sets and tilings, this is the case by construction, but for inflation tilings the situation is not as clear It is known (Solomyak 1997) that a self-similar inflation tiling has a non-trivial pure point component in its spectrum if, and only if, the scaling factor of the inflation is a Pisot number λ; that is, a real algebraic integer λ > 1 all of whose conjugates are strictly smaller than one in modulus. In the case of a canonical (in particular, irrational and aperiodic) cut-and-project tiling, it is known (Julien 2010, Theorem 5.1) that the complexity function grows like O(nd ) if, and only if, the groups of cohomology over Q of the tiling space are finitely generated. It is concerned with the speed of convergence of the number of points of a Delone set inside larger and larger balls

Properties of Patterns and Properties of Tiling Spaces

A “Folklore” Problem Concerning the Properties of Cut-and-Project Sets