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](https://en.wikipedia.org/wiki/Schnirelmann_density) 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 $S^2$.



Additive basis of a set union the square of the set
In the paper "On additive bases. II", Deshouillers and Fouvry prove a conjecture (made in part I, by a different set of authors) that for each sequence $K$ of positive integers, there is a set $A$ such that $A^k$ is a basis precisely when $k$ belongs to $K$. See
J. London Math. Soc. (2) 14 (1976), no. 3, 413–422.


In particular, one can have $A$ not a basis but $A^2$ is a basis (so that $A\cup A^2$ is also a basis). This case in fact follows from what is done in Part I.


