Let us define the **boxing dimension** $\text{bim}(S)$ of a set $S \subseteq \mathbb Z$ as the smallest integer $k \ge 0$ for which there is a family of $k$ **discrete intervals** (that is, finite or infinite intervals of the poset of integers with their usual ordering) that cover $S$ (in the sense that $S$ is <strike>contained in</strike> equal to their union), with the understanding that if no such integer $k$ exists then $\text{bim}(S) := \infty$.

The boxing dimension of the set $S$ is zero if and only if $S$ is empty, and it is one if and only if $S$ is a discrete interval. More generally, the boxing dimension of $S$ is equal to a certain integer $k \ge 0$ if and only if there is a *unique* way to decompose $S$ as a union of $k$ well-separated non-empty discrete intervals, where we say that two sets $X, Y \subseteq \mathbb Z$ are **well separated** if $|x-y| \ge 2$ for all $x \in X$ and $y \in Y$ (so, well-separated sets are disjoint).

For instance, the sets $A := \{0, 5\} \cup \mathbb N_{\ge 7}$ and $B := \{-2, 2, 3\}$ are well separated, with $\text{bim}(A) = 3$ and $\text{bim}(B) = 2$. To the contrary, the set of even integers and the set of odd integers are *not* well separated, with the boxing dimension of each of them being infinite.

**MY QUESTIONS.** (1) Does (what I'm calling) the boxing dimension have a more standard name? Or is it a special case of a more general *standard* notion? (2) Likewise, do well-separated subsets of $\mathbb Z$ have a more standard name? Or are they a special case of a more general *standard* notion?

For what it's worth, my interest in this notion comes from a joint project with Weihao Yan (an undergraduate student at Hebei Normal University), where the classification of the automorphism group of certain objects boils down to a (somewhat unusual) induction on the boxing dimension of certain finite subsets of $\mathbb N$.