Some properties of sets in the plane closed under affine combination by a fixed parameter

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.