Subsets of $\mathbb Z/n\mathbb Z$ that remain disjoint with themselves under shifts

Question

Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ such that for any $a\ne 0$ in $\mathbb Z/n\mathbb Z$, $X$ is disjoint with $X + a = \{x + a \pmod n\mid x \in X\}$?

Answer 1

If a subset $X$ of a group $G$ is disjoint with $aX = \{ax: x \in X\}$ for any $a \in G-\{e\}$, then $X$ is either empty or one-element. Suppose, $\exists x, y \in X x \neq y$, then $xy^{-1} \neq e$ and $x \in xy^{-1}X \cap X$. However, the empty subset and all one element subsets clearly satisfy your condition.