Properties of two-dimensional sets with small sumset

Let A, B ⊆ R be finite, nonempty subsets, let s ≥ 2 be an integer, and let h1(A,B) denote the minimal number t such that there exist 2t (not necessarily distinct) parallel lines, l1, . . . , lt, l ′ 1, . . . , l ′ t , with A ⊆ ⋃t i=1 li and B ⊆ ⋃t i=1 l ′ i . Suppose h1(A,B) ≥ s. Then we show that: (a) if ||A| − |B|| ≤ s and |A|+ |B| ≥ 4s − 6s+ 3, then |A+B| ≥ (2− 1 s )(|A| + |B|)− 2s+ 1; (b) if |A| ≥ |B|+ s and |B| ≥ 2s − 72s+ 3 2 , then |A+B| ≥ |A|+ (3− 2 s )|B| − s; (c) if |A| ≥ 1 2s(s−1)|B|+s and either |A| > 18 (2s−1)2|B|− 1 4 (2s−1)+ (s−1) 2(|B|−2) or |B| ≥ 2s+4 3 , then |A+B| ≥ |A|+ s(|B| − 1). This extends the 2-dimensional case of the Freiman 2–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 2-dimensional case of estimates of Green and Tao and of Gardner and Gronchi, and that generalize the 2-dimensional case of the Brunn-Minkowski Theorem.