Unbounded Cantor like set.

Question

Consider the interval $[0,\infty)$,if we construct a subset of this unbounded closed interval as follows, $A=\bigcup_{n\in \mathbb N}$$(n+C)$, where $C$ denotes Cantor set and $n+C$ is the translation of Cantor set by $n$.Then this set is nowhere dense because it has similar structure as Cantor set,it is unbounded above.Every point of it,is a condensation point of the set and the set is closed.Are there some more interesting properties of this set?