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 subset of R-2 in terms of the minimum number h, (A, B) of parallel lines covering each of A and B. We show that, if h(1) (A, B) >= s and vertical bar A vertical bar >= vertical bar B vertical bar >= 2s(2) - 3s + 2, then vertical bar A + B vertical bar >= vertical bar A vertical bar + (3 - 2/s)vertical bar B vertical bar - 2s + 1. More precise estimations are given under different assumptions on vertical bar A vertical bar and vertical bar B vertical bar. This extends 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 vertical bar A vertical bar = vertical bar B vertical bar (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. (C) 2009 Elsevier Inc. All rights reserved.