Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ with the following property: there exists $a\ne 0$ in $\mathbb Z/n\mathbb Z$ such that $X$ is disjoint with $X + a = \{x + a \pmod n\mid x \in X\}$?
For example, for $n=5$, any set with 1 or 2 elements satisfies this property. I used to believe that the same holds for any subsets of size less than or equal to $\frac{n}{2}$, but for $n=6$, a counterexample can be easily constructed.
Probably there’s some well-known theorem about it?
Denote by $X-X=\{x-y\mid x,y\in X\}$.
Then: $X$ is disjoint from some of its shift iff $X-X\subsetneq\mathbb Z/n\mathbb Z$. Moreover:
For $(\Rightarrow)$: if $X\cap (a+X)=\emptyset$, then $a\notin X-X$, because otherwise $a=x-y$ for some $x,y\in X$, but then $x=a+y\in X\cap (a+X)$.
For $(\Leftarrow)$: if $X\cap (a+X)\neq\emptyset$, then $x=a+y$ for some $x,y\in X$, so $a=x-y\in X-X$. $\square$
Note that something similar holds for any group $G$. For $X\subseteq G$, $X\cap aX=\emptyset$ iff $a\notin XX^{-1}$, and $X\cap Xa=\emptyset$ iff $a\notin X^{-1}X$. (Here $XX^{-1}=\{xy^{-1}\mid x,y\in X\}$, and similarly for $X^{-1}X$.)