Sumsets in the set of squares

We study sumsets $\mathcal{A}+\mathcal{B}$ in the set of squares $\mathcal{S}$ (and, more generally, in the set of kth powers $\mathcal{S}_k$, where $k\geq 2$ is an integer). It is known by a result of Gyarmati that $\mathcal{A}+\mathcal{B}\subset \mathcal{S}_k \cap [1,N]$ implies that $\min(|\mathcal{A}|,|\mathcal{B}|)=O_k(\log N)$. Here, we study how the upper bound on $|\mathcal{B}|$ decreases, when the size of $|\mathcal{A}|$ increases (or vice versa). In particular, if $|\mathcal{A}|\geq C k^{\frac{1}{m}} m (\log N)^{\frac{1}{m}}$, then $|\mathcal{B}|=O_k(m^2 \log N)$, for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with $m\sim \log \log N$ this gives: If $|\mathcal{A}|\geq C_k \log \log N$, then $|\mathcal{B}|=O_k(\log N (\log \log N)^2)$.