Sumsets in the set of squares

We study sumsets A+B in the set of squares S (and, more generally, in the set of kth powers Sₖ, where k≥ 2 is an integer). It is known by a result of Gyarmati that A+B⊂ Sₖ ∩ [1,N] implies that min(|A|,|B|)=Oₖ(log N). Here, we study how the upper bound on |B| decreases, when the size of |A| increases (or vice versa). In particular, if |A|≥ C k1/m m (log N)1/m, then |B|=Oₖ(m² log N), for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with m~ log log N this gives: If |A|≥ Cₖ log log N, then |B|=Oₖ(log N (log log N)²).