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It is well known that size of the set of positive integers up to $n$ that can be written as $a^2+b^2$ is asymptotic to $C \frac{n}{\sqrt{\log n}}$. Here I'm interested mostly in the weaker fact that ...

Recall that if $A$ and $B$ are both subsets of the integers, then $A+B=\{a+b:a \in A,b \in B\}$. Lagrange's four-square theorem states that if $A$ is the set of squares, then $4A=A+A+A+A=\mathbb{N}$. ...