For $E\subseteq\mathbb{R}$, let us define the “difference set” of $E$ as:$$\Delta E := \{x-y : (x,y)\in E^2\}$$Furthermore, when $\mathcal{C} \subseteq \mathcal{P}(\mathbb{R})$, let$$\Delta[\mathcal{C}] := \{\Delta E : E\in \mathcal{C}\}$$Now let $\mathcal{D}^0$ be the set of discrete subsets of $\mathbb{R}$ (where, to be completely clear, “$S$ is discrete” means $S$ inherits the discrete topology from $\mathbb{R}$, i.e., for every $x\in S$ there exists $\varepsilon>0$ such that $x$ is the only element of $S$ between $x-\varepsilon$ and $x+\varepsilon$).
Question: can we characterize the difference sets of discrete sets? I.e., what is $\mathcal{D}^1 := \Delta[\mathcal{D}^0]$ with the above notation? (Obviously they are countable, because discrete sets are countable and $\Delta$ takes a countable set to a countable set. But do we get all countable sets like this?)
If we don't get all countable sets as $\Delta[\mathcal{D}^0]$ then we can further iterate: define $\mathcal{D}^\alpha$ (for $\alpha$ ordinal) by$$\begin{aligned}\mathcal{D}^{\alpha+1} &:= \Delta[\mathcal{D}^\alpha]\\\mathcal{D}^{\delta} &:= \bigcup_{\xi<\delta} \mathcal{D}^\xi\;\text{ for $\delta$ limit}\end{aligned}$$Again, by straightforward induction, all elements of all the $\mathcal{D}^\alpha$ are countable. The family $\mathcal{D}^\alpha$ must stabilize (before $2^{\aleph_0}$) to some $\mathcal{D}^{\infty} := \bigcup_{\alpha} \mathcal{D}^\alpha$ (viꝫ. the smallest set of subsets of $\mathbb{R}$ containing the discrete sets and closed under taking difference sets):
Further question: What is this $\mathcal{D}^{\infty}$ and what is the smallest $\alpha$ such that $\mathcal{D}^{\alpha+1} = \mathcal{D}^\alpha$ then?