Origin and exposition $\mathbb N$ can be partitioned into infinitely many subsets $A+b_k$ for some infinite $A$ I came across this fantastic mathematical result and I can't help but think that it's too amazing a result to not have a paper on it or at least be named after somebody ! Unfortunately, the book just presented it as a dull commonplace paragraph that students should be able to solve routinely.
The question is : Is there a way to partition the positive integers into infinitely many infinite subsets such that each subset is generated from any other by adding the same positive integer to each element of the subset. 
The proof is strikingly simple.
First we have two sets, $A$ and $B$. $A$ consists of all numbers with zeroes at odd positions and $B$ consists of all numbers with zeroes at even positions. Every positive integer can be uniquely represented as $n = a + b$,$ a \in A, b \in B$. Now, the infinite family of sets are made as follows : $A_1 = A$, $A_k = A + b_k$, where $b_k$ is the $k$-th element of $B$.
Also, if someone could offer further exposition on the proof it would be great. I don't think this proves ANY subset can be generated from any other, just proves that any subset can be generated from one subset.
Now, I want to know who first found this result or if it's really just a commonplace result with nothing special about it
 A: The result isn’t quite true the way you’ve stated it. What is true is that we can partition $\Bbb Z^+$ into infinite sets $A_k$ for $k\in\Bbb Z^+$ so that whenever $k,\ell\in\Bbb Z^+$ and $k\ne\ell$, there is a positive integer $n$ such that either $A_k=A_\ell+n$ or $A_\ell=A_k+n$. Equivalently, whenever $k,\ell\in\Bbb Z^+$, there is an integer $n$, not necessarily positive, such that $A_\ell=A_k+n$ (and hence $A_k=A_\ell-n$).
The sets $A_k$ defined in your question do indeed satisfy this statement. Suppose that $k$ and $\ell$ are distinct positive integers; without loss of generality we may assume that $b_k<b_\ell$. Then
$$\begin{align*}
A_\ell&=A+b_\ell\\
&=A+b_k+(b_\ell-b_k)\\
&=A_k+(b_\ell-b_k)\;,
\end{align*}$$
where clearly $b_\ell-b_k\in\Bbb Z^+$. In particular, if we enumerate $B=\{b_k:k\in\Bbb Z^+\}$ in increasing order, so that $b_k<b_\ell$ whenever $k<\ell$, then we have $A_\ell=A_k+(b_\ell-b_k)$ whenever $k<\ell$.
I grant that the existence of such a partition is at least a little surprising, and it does take a bit of ingenuity (or luck!) to come up with one, but I suspect that this is what mathematicians call a folklore result: something that is widely known without being certainly attributable to anyone.
