s-convexity, model sets and their relation

TL;DR

This paper explores the connection between s-convexity and model sets for physical quasicrystals, highlighting special cases linked to specific Pisot numbers and their implications for non-crystallographic symmetries.

Contribution

It characterizes 1-dimensional model sets using s-convexity and identifies exceptional Pisot numbers related to observed quasicrystal symmetries.

Findings

1-dimensional model sets can be characterized by s-convexity for finite s

Certain Pisot numbers are exceptional in s-convexity context

Specific Pisot numbers relate to observed non-crystallographic symmetries

Abstract

The relation of s-convexity and sets modeling physical quasicrystals is explained for quasicrystals related to quadratic unitary Pisot numbers. We show that 1-dimensional model sets may be characterized by s-convexity for finite set of parameters s. It is shown that the three Pisot numbers $\frac12(1+\sqrt5)$, $1+\sqrt2$, and $2+\sqrt3$ related to experimentally observed non-crystallographic symmetries are exceptional with respect to s-convexity.