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 ⊆ R 2 in terms of the minimum number h 1 ( A , B ) of parallel lines covering each of A and B. We show that, if h 1 ( A , B ) ⩾ s and | A | ⩾ | B | ⩾ 2 s 2 − 3 s + 2 , then | A + B | ⩾ | A | + ( 3 − 2 s ) | B | − 2 s + 1 . More precise estimations are given under different assumptions on | A | and | B | . xtends the 2-dimensional case of the Freiman 2 d -Theorem to distinct sets A and B, and, in the symmetric case A = B , improves the best prior known bound for | A | = | B | (due to Stanchescu, and which was cubic in s) to an exact value. t 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.