We have a similar concept on numerical semigroups, borrowed from the idea of [run](https://link.springer.com/referenceworkentry/10.1007/0-387-23483-7_178).

For a numerical semigroup, a [desert](https://gap-packages.github.io/numericalsgps/doc/chap3.html) is an run of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the [second Feng-Rao number of the semigroup](https://ieeexplore.ieee.org/document/6655888), related to algebro-geometric codes. 

In this setting, for a numerical semigroup $S$, then you can count deserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$. 

In any case, I my feeling is that you should allow infinite intervals in your definition. Then, the Apéry set of 1 would also recover the interval $[-\infty,-1]$.