Recall a set of integers $S$ is said to be an additive basis for the natural numbers if there is a $k$ such that every positive integer is expressible as a sum of at most $k$ elements of $S$. Similarly, a set $S$ is said to be an asymptotic additive basis for the natural numbers if there is a $k$ such that every sufficiently large integer is the sum of at most $k$ (not necessarily distinct) elements of $S$.  Lagrange's four-square theorem can be thought of as a statement that the squares are an additive basis with $k=4$. 

Given a set $S$, we will write $S^2= \{s^2: s \in S\}$. 

**Question:** Is there an example of a set $S$ which is not an additive basis but where $S \cup S^2$ is an additive basis? (The same question then for asymptotic additive basis but I will not focus on that here.)

Note that any set with positive [Schnirelmann density][1] is an additive basis, so one naive way of solving this would be to exhibit a set $S$ which is not an additive basis but where $S \cup S^2$ has positive Schnirelmann density but this does not work; if $S$ has Schnirelmann density density zero then so will $S^2$.

  [1]: https://en.wikipedia.org/wiki/Schnirelmann_density