Shifting Finite Sets to Cover Points without Intersection

Question

I am interested in finite subsets $S\subseteq\mathbb{R}$ that may be shifted to cover any point not contained within themselves on the real line without overlap. Formally, these are sets $S\subseteq\mathbb{R}$ such that for each $r\in\mathbb{R}\setminus S$, there exists $a\in\mathbb{R}$ such that $r\in S+a$, and $S\cap S+a=\emptyset$, where for a set $X\subseteq\mathbb{R}$ and $b\in\mathbb{R}$, $X+b$ is defined to be $\{x+b\ |\ x\in X\}$. Equivalently, for all $r\in\mathbb{R}\setminus S$ there exists $s\in S$ such that $r-s\not\in S-S$, where $X-X$ for $X\subseteq\mathbb{R}$ is defined to be $\{x-y\ |\ x,y\in X\}$. Examples include every set with one or two members, and every set with three members $a<b<c$ such that $b-a$ is neither half nor twice $c-b$ (the images of $\{0,1,3\}$ under injective affine transformations). Is there a known characterisation of these sets?