Properties of two-dimensional sets with small sumset

We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B⊆R2A,B⊆R2 in terms of the minimum number h1(A,B)h1(A,B) of parallel lines covering each of A and B  . We show that, if h1(A,B)⩾sh1(A,B)⩾s and |A|⩾|B|⩾2s2−3s+2|A|⩾|B|⩾2s2−3s+2, then|A+B|⩾|A|+(3−2s)|B|−2s+1. More precise estimations are given under different assumptions on |A||A| and |B||B|.This extends the 2-dimensional case of the Freiman d22d-Theorem to distinct sets A and B  , and, in the symmetric case A=BA=B, improves the best prior known bound for |A|=|B||A|=|B| (due to Stanchescu, and which was cubic in s) to an exact value.As part of the proof, we give general lower bounds for two-dimensional subsets that improve the two-dimensional case of estimates of Green and Tao and of Gardner and Gronchi, related to the Brunn–Minkowski Theorem.