What are the pairs $(P,Q)$ of subsets of $\mathbb N$ for which the map \begin{eqnarray*} P\times Q & \rightarrow & {\mathbb N} \\\\ (p,q) & \mapsto & p+q \end{eqnarray*} is a bijection ?

Obvious examples are $P=\mathbb N$ with $Q=\{0\}$, or $P=2\mathbb N$ with $Q=\{0,1\}$. Are there others ?

This question is related to a puzzle given in EMISSARY (fall 2010), asking to find infinite series $f(x)$ and $g(x)$ with coefficients $0$ and $1$, whose product equals $\frac{1}{1-x}$. I suspect that the word *infinite* was written on purpose, and therefore $P$ and $Q$ must be infinite.

**Later**. After the answers, I understand that one can find a sequence $(P_j)_{j\ge0}$ of subsets of $\mathbb N$ with $0\in P_j$, such that every $n\in\mathbb N$ writes $\sum_{j\ge0}p_j$ with $p_j\in P_j$, in a unique way. Of course, all but finitely many $p_j$'s are zeros. Now, I feel dumb, because this follows for instance from the writing of integers in some basis.

`$$\begin{eqnarray*} P\times Q & \rightarrow & {\mathbb Z} \\ (p,q) & \mapsto & p+q \end{eqnarray*}$$`

can be quite bizarre. $\endgroup$ – Aaron Meyerowitz Jan 1 '11 at 8:53