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.