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 \( \left| {\mathcal{K} + \mathcal{K}} \right| \).

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Y. Bilu: Structure of sets with small sumsets, Astérisque 258 (1999), 77–108.

    MathSciNet  Google Scholar 

  2. [2]

    M. Chang: A polynomial bound in Freiman’s theorem, Duke Math. J. 113 (2002), 399–419.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    G. A. Freiman: Foundations of a Structural Theory of Set Addition, Transl. of Math. Monographs, 37, A.M.S., Providence, R.I., 1973.

    Google Scholar 

  4. [4]

    G. A. Freiman: Structure Theory of Set Addition, Astérisque 258 (1999), 1–33.

    MathSciNet  Google Scholar 

  5. [5]

    G. A. Freiman: Structure Theory of Set Addition, II. Results and Problems; in: P. Erdős and his Mathematics, I, Budapest, 2002, pp. 243–260.

  6. [6]

    I. Z. Ruzsa: Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379–388.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    Y. V. Stanchescu: On the structure of sets of lattice points in the plane with a small doubling property, Astérisque 258 (1999), 217–240.

    MathSciNet  Google Scholar 

  8. [8]

    Y. V. Stanchescu: On the structure of sets with small doubling property on the plane (I), Acta Arithmetica LXXXIII.2 (1998), 127–141.

    MathSciNet  Google Scholar 

  9. [9]

    Y. V. Stanchescu: On the simplest inverse problem for sums of sets in several dimensions, Combinatorica 18(1) (1998), 139–149.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yonutz V. Stanchescu.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Mathematics Subject Classification (2000)

  • 11P70
  • 11B25
  • 52C99