## Fall 2017, problem 54

Recall that a sum of two sets $A$ and $B$ is defined as $$ A + B = \left\{ a + b | a \in A, b \in B \right\}. $$ Can one find three sets $A$, $B$, and $C$ such that $A \cup B \cup C = \mathbb{Z}$, the sets are pairwise disjoint, and the sets $A+B$, $A+C$, and $B+C$ are pairwise disjoint? Can one do this if you replace $\mathbb{Z}$ with $\mathbb{Q}$?

An example in $\mathbb Z$ is the following one.

Set $A:=\{3z|z\in \mathbb Z\}, B:=\{3z+1|z\in\mathbb Z\}, C:=\{3z+2|z\in\mathbb Z\}$.

Then A∪B∪C=$\mathbb Z$; the sets are pairwise disjoint, as well as their sums (because A+B=B, A+C=C, B+C=A).