Fixed-Parameter Extrapolation and Aperiodic Order

TL;DR

This paper studies the properties of $ ext{lambda}$-convex and $ ext{lambda}$-clonvex sets in the complex plane, characterizing when they are convex or discrete, and exploring their connections to quasicrystals and aperiodic order.

Contribution

It introduces new conditions for convexity and discreteness of $ ext{lambda}$-convex sets, linking algebraic properties of $ ext{lambda}$ to geometric and aperiodic structures.

Findings

$R_ ext{lambda}$ is either convex or uniformly discrete.

The set of $ ext{lambda}$ values making $R_ ext{lambda}$ convex is characterized.

Connections between $ ext{lambda}$-convex sets and quasicrystals are established.

Abstract

Fix any $\lambda\in\mathbb{C}$. We say that a set $S\subseteq\mathbb{C}$ is $\lambda$-$convex$ if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that $S$ is $\lambda$-$clonvex$. We investigate the properties of $\lambda$-convex and $\lambda$-clonvex sets and prove a number of facts about them. Letting $R_\lambda\subseteq\mathbb{C}$ be the least $\lambda$-clonvex superset of $\{0,1\}$, we show that if $R_\lambda$ is convex in the usual sense, then $R_\lambda$ must be either $[0,1]$ or $\mathbb{R}$ or $\mathbb{C}$, depending on $\lambda$. We investigate which $\lambda$ make $R_\lambda$ convex, derive a number of conditions equivalent to $R_\lambda$ being convex, and give several conditions sufficient for $R_\lambda$ to be convex or not convex; in particular, we show that $R_\lambda$ is either convex or uniformly discrete. Letting $\mathcal{C} := \{\lambda\in\mathbb{C}\mid \mbox{$R_\lambda$ is convex}\}$, we show that $\mathbb{C}\setminus\mathcal{C}$ is closed, discrete and contains only algebraic integers. We also give a sufficient condition on $\lambda$ for $R_\lambda$ and some other related $\lambda$-convex sets to be discrete by introducing the notion of a strong PV number. These conditions give rise to a number of periodic and aperiodic Meyer sets (the latter sometimes known as "quasicrystals"). The paper is in four parts. Part I describes basic properties of $\lambda$-convex and $\lambda$-clonvex sets, including convexity versus uniform discreteness. Part II explores the connections between $\lambda$-convex sets and quasicrystals and displays a number of such sets, including several with dihedral symmetry. Part III generalizes a result from Part I about the $\lambda$-convex closure of a path, and Part IV contains our conclusions and open problems.