Three-dimensional sets with small sumset

Abstract

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let $$\mathcal{K}$$ be a finite subset of ℤ3 such that dim $$\mathcal{K}$$ = 3. If $$\left| {\mathcal{K} + \mathcal{K}} \right| 12^3$$, then $$\mathcal{K}$$ lies on three parallel lines. Moreover, for every three dimensional finite set $$\mathcal{K} \subseteq \mathbb{Z}^3$$ that lies on three parallel lines, if $$\left| {\mathcal{K} + \mathcal{K}} \right| < 5\left| \mathcal{K} \right| - 10$$, then $$\mathcal{K}$$ is contained in three arithmetic progressions with the same common difference, having together no more than $$\upsilon = \left| {\mathcal{K} + \mathcal{K}} \right| - 3\left| \mathcal{K} \right| + 6$$ terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for $$\left| {\mathcal{K} + \mathcal{K}} \right|$$.