Fixed-Parameter Extrapolation and Aperiodic Order.

Fix any $lambdainmathbb{C}$. We say that a set $Ssubseteqmathbb{C}$ 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 $R_lambdasubseteqmathbb{C}$ 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 $mathcal{C} := {lambdainmathbb{C}mid mbox{$R_lambda$ is convex}}$, we show that $mathcal{C}$ is open and contains all transcendental $lambda$. It follows that $mathbb{C} setminus mathcal{C}$ is countable, closed, and nowhere dense. 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 ``quasicrystalsu0027u0027). 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 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, and Part IV contains our conclusions and open problems.