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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References
Y. Bilu: Structure of sets with small sumsets, Astérisque 258 (1999), 77–108.
M. Chang: A polynomial bound in Freiman’s theorem, Duke Math. J. 113 (2002), 399–419.
G. A. Freiman: Foundations of a Structural Theory of Set Addition, Transl. of Math. Monographs, 37, A.M.S., Providence, R.I., 1973.
G. A. Freiman: Structure Theory of Set Addition, Astérisque 258 (1999), 1–33.
G. A. Freiman: Structure Theory of Set Addition, II. Results and Problems; in: P. Erdős and his Mathematics, I, Budapest, 2002, pp. 243–260.
I. Z. Ruzsa: Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379–388.
Y. V. Stanchescu: On the structure of sets of lattice points in the plane with a small doubling property, Astérisque 258 (1999), 217–240.
Y. V. Stanchescu: On the structure of sets with small doubling property on the plane (I), Acta Arithmetica LXXXIII.2 (1998), 127–141.
Y. V. Stanchescu: On the simplest inverse problem for sums of sets in several dimensions, Combinatorica 18(1) (1998), 139–149.