Sumsets in the set of squares

Abstract 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 (|A|, |B|) =Oₖ (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 k^1{m} m (N) ^1{m}, then |B|=Oₖ (m² N), for sufficiently large N, a positive integer m and an explicit constant C 0. For example, with m N this gives: If |A| Cₖ N, then |B|=Oₖ (N (N) ²).