## 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 ℤ^{3} 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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Stanchescu, Y.V. Three-dimensional sets with small sumset.
*Combinatorica* **28, **343–355 (2008). https://doi.org/10.1007/s00493-008-2205-4

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### Mathematics Subject Classification (2000)

- 11P70
- 11B25
- 52C99