Properties of two-dimensional sets with small sumset

摘要

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 - frac(2, s)) | B | - 2 s + 1. More precise estimations are given under different assumptions on | A | and | B |. 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 | 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.