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 $\left| {\mathcal{K} + \mathcal{K}} \right| < \tfrac{{13}} {3}\left| \mathcal{K} \right| - \tfrac{{25}} {3}$ and $\left| {\mathcal{K} + \mathcal{K}} \right| > 12^3$, 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 $\left| {\mathcal{K} + \mathcal{K}} \right|$.