Planar subsets with small doubling

Abstract

Let A, B ⊆ ℤ2 be finite, non-empty subsets each covered by precisely two horizontal lines. Suppose (A + B - A - B) = ℤ2, |A| ≥ |B| and |A + B| = |A| + 2|B| - 3 + r ≤ |A| + 19/7|B| - 5. Then there exist subsets PA,PB,P ⊆ ℤ2, each the union of two arithmetic progressions with difference (1,0), such that A ⊆ PA, B ⊆ PB and (x + A) ∪ (y + B) ⊆ P, for some x,y ∈ ℤ2, with |PA| ≤ |A| + r, |PB| ≤ |B| + r, |PA| + |PB| ≤ 2|B| + 2r and |P| ≤ (|A| + |B|)/2 +3/2r. A similar result is proved assuming that A is covered by two horizontal lines and B by one and vice versa. This generalizes a result of Stanchescu handling the case A = B and extends the Freiman 3k - 4 Theorem to two-dimensional sumsets with |A + B| < |A| + 7/3 |B| - 5.

Publication Title

Quarterly Journal of Mathematics

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