I was trying to partition $\mathbb R$ into two sets $A, B$ such that for all $a\in A, b\in B$ we have $|a-b|\neq 1$. An obvious way to do it is to take $\mathbb Z$ and ${\mathbb R}\setminus {\mathbb Z}$. The other examples I found all consisted of one countable set and its complement.

Question. Is there $A\subseteq {\mathbb R}$ such that both $A$ and $B:= {\mathbb R}\setminus A$ are uncountable, and for all $a\in A, b\in B$ we have $|a-b|\neq 1$?