Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}\_{p}$.


Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity:


$$\max (|\{a+b : a,b \in A\}|, |\{a^2+b^2 : a,b \in A\}| ) ? $$


In other words, is either the sumset of $A$ or the sumset of the square set of $A$ guaranteed to be large? 


This is very similar to the sum-product problem (which is formally connected to the variant question of lower bounding $\max(|2A|, |2A^2|)$). My hope is that this problem might be easier than the sum-product problem and better bounds may be available.



Is the sumset or the sumset of the square set always large?
A more general result than what you want appears as Theorem 1 in <http://arxiv.org/pdf/1002.2554>. (A slightly weaker result had appeared before as Theorem 3.1 in <http://arxiv.org/pdf/0909.5471>).


Curiously, it is open if at least one of $A^2+A^2$ and $A^3+A^3$ is necessarily large.


