Is the sumset or the sumset of the square set always large? 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.
 A: 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.
A: Six years later it's hard to know if this is still a question of interest. For the record (since I stumbled across the question), the answer is yes and it's a consequence of a more general phenomenon.
The main theorem of the paper of Stevens and Warren (Thm 5 in https://arxiv.org/abs/2102.05446) states that for any convex (or concave) functions $f, g:\mathbb{R}\rightarrow \mathbb{R}$ we have
$$
|A+B|^{38}|f(A)+g(B)|^{38} \gtrsim |A|^{49}|B|^{49}
$$
where $\gtrsim$ hides factors of $\log(|A|)$.
Then the answer to the original question follows by setting $A = B$ and $f = g$ with $f(x) = x^2$. This gives, with $S = \{a^2 : a \in A\}$
$$
\max\{|A+A|,|S+S|\} \gtrsim |A|^{49/38}
$$
i.e. an exponent of $1.28947\ldots$
A: I believe I can make some progress in the real case using a variant of Elekes' approach to the sum-product problem based on the Szemeredi-Trotter theorem for unit paraboloids.
Let $A$ be a finite set of real numbers and $S = \{a^2 : a \in A\}$.
Consider the subset $B \subset \mathbb{R}^2$ given by
$$(a+b, c^2+d^2 : a,b,c,d \in A).$$
Clearly $|B| = |A+A| \times |S+S|  $.
Now for a given pair $u,v \in \mathbb{R}$ define the paraboloid $p_{(u,v)}$ to be the set of solutions to the equation
$$y= (x-u)^2 + v^2.$$
Now each such paraboloid will contain at least $|A|$ points from $B$, namely the points of the form
$$ (u+a, a^2+v^2)$$
where $a$ varies over $A$ and $u$ and $v$ are fixed (depending on $p_{(u,v)}$). Let $L=\{p_{(u,v)} : u,v \in A \} $ denote the set of parabolas generate as described above, we must have
$$|I(B,L)|  \geq |A|^3  $$
On the other hand, by the Szemeredi-Trotter theorem (for "unit" paraboloids), we have
$$|A|^3 \leq |I(B,L)| \lesssim |A+A|^{2/3} |S+S|^{2/3} |A|^{4/3} + |A+A||S+S| + |A|^2$$
which implies $|A|^{5/2}\lesssim |A+A| \times |S+S|$. This, in turn, implies that
$$|A|^{5/4} \leq \max(|A+A|,|S+S|).$$
