Paradoxical decompositions of sets usually require the axiom of choice; Hausdorff or Banach-Tarski are well-known examples. A paradoxical decomposition of a point set without the axiom of choice has been constructed by Sierpinski and Mazurkiewicz. A set $S$ is the union of two sets $A$ and $B$. When the elements of $A$ are rotated ($\rho$) by one radian, then $\rho$$A = S$, and when the elements of $B$ are translated ($\tau$) by one unit, then $\tau$$B = S$ too.

There is a simple variant. Decompose the set $\mathbb{Z}$ of all integers into $A$, the set of even integers, and $B$, the set of odd integers. When the elements of $A$ are divided $(\delta)$ by 2, then $\delta$$A = \mathbb{Z}$. When the elements of $B$ are translated by one unit (in positive or negative direction) and then divided by 2, then $\delta\tau$$B = \mathbb{Z}$.

Same can be shown for other sets $S$, for instance the set of positive integers (then $B$ must be translated by +1).

My question: Have these paradoxical decompositions already appeared in literature? I would like to include them into my lectures with appropriate quotation but could not yet find a source or an author.