# 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 $$\left| {\mathcal{K} + \mathcal{K}} \right|$$.

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Correspondence to Yonutz V. Stanchescu.

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Stanchescu, Y.V. Three-dimensional sets with small sumset. Combinatorica 28, 343–355 (2008). https://doi.org/10.1007/s00493-008-2205-4