A diophantine characterization of linear repetitivity in cut and project sets: quantitative results.

21.06.2023, 13:00  –  Haus 9, Raum 0.17 und Zoom
Forschungsseminar Diskrete Spektraltheorie

Johannes Happich (Leipzig)

Abstract: When comparing the complexity of different aperiodic quasicrystals, it appears that linear repetitivity is a useful property that only applies to the - in some sense - most regular quasicrystals. It is thus desirable to study different ways of describing this property. We will introduce a result by Haynes, Koivusalo and Walton, who proved a characterization of linear repetitivity in cubical cut and project sets by some bad approximability conditions. Furthermore, we present new explicit bounds on the involved constants in some special cases. We give a brief account of all relevant notions  on Delone sets,  cut and project sets and diophantine approximation. After stating the results,  we proceed by presenting some methods of proof. Finally, we investigate the further potential of these methods and formulate possible generalisation. If time permits, we will also talk about consequences of the theory for notions of entropy in cut and project sets.