An improved construction of progression-free sets

  • Michael ElkinEmail author


The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with \(|S| = \Omega ({n^{{{\log }_3}2}})\) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is
$$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }} \cdot {{\log }^{1/4}}n}}} \right).$$
Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend has been reported since 1946.
In this paper we present a construction that improves the result of Behrend by a factor of \({\rm{\Theta }}\left( {\sqrt {\log n} } \right)\) , and shows that
$$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }}}} \cdot {{\log }^{1/4}}n} \right).$$
In particular, our result implies that the construction of Behrend is not optimal.

Our construction and proof are elementary and self-contained. We also present an application of our proof technique in Discrete Geometry.


  1. [1]
    H. L. Abbott, On a conjecture of Erdős and Straus on non-averaging sets of integers, in Proceedings of the 5th British Combinatorial Conference, Congress Numerartium XV, Utilitas Mathematica, Winnipeg, Man., 1975, pp. 1–4.Google Scholar
  2. [2]
    H. L. Abbott, Extremal problems on non-averaging and non-dividing sets, Pacific Journal of Mathematics 91 (1980), 1–12.MathSciNetzbMATHGoogle Scholar
  3. [3]
    H. L. Abbott, On the Erdős-Straus non-averaging set problem, Acta Mathematica Hungarica 47 (1986), 117–119.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    S. D. Adhikari, Lattice points in spheres, Bulletin of the Allahabad Mathematical Society 8–9 (1993–1994), 1–13.MathSciNetGoogle Scholar
  5. [5]
    G. E. Andrews, A lower bound for the volumes of strictly convex bodies with many boundary points, Transactions of the American Mathematical Society 106 (1963), 270–279.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    V. I. Arnold, Statistics of integer convex polytopes (in Russian), Funk. Anal. Pril. 14 (1980), no. 1, 1–3.Google Scholar
  7. [7]
    A. Balog and I. Barany, On the convex hull of the integer points in a disc, DIMACS Series on Discrete and Computational Geometry 6 (1991), 39–44.MathSciNetGoogle Scholar
  8. [8]
    I. Barany and D. G. Larman, The convex hull of the integer points in a large ball, Mathematische Annalen 312 (1998), 167–181.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    F. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proceedings of the National Academy of Sciences U.S.A 32 (1946), 331–332.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. P. Bosznay, On the lower estimation of non-averaging sets, Acta Mathematica Hungarica 53 (1989), 155–157.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    J. Bourgain, On triples in arithmetic progression, Geometric and Functional Analysis 9 (1999), 968–984.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    J. Bourgain, Roth’s theorem in progressions revisited, Journal D’Analise Mathematique 104 (2008), 155–192.MathSciNetzbMATHGoogle Scholar
  13. [13]
    D. Coppersmith, Personal communication, 2003.Google Scholar
  14. [14]
    M. Elkin, An improved construction of progression-free sets, in ACM-SIAM Symposium on Discrete Algorithms, SODA’10, Austin, TX, USA, 2010, pp. 886–905.Google Scholar
  15. [15]
    P. Erdős and P. Turán, On some sequences of integers, Journal of the London Mathematical Society 11 (1936), 261–264.CrossRefGoogle Scholar
  16. [16]
    G. A. Freiman, Inverse problems of additive number theory, Izvestiya Akademii Nauk SSSR, (in Russian) Seriya Matematicheskaya 19 (1955), 275–284.MathSciNetGoogle Scholar
  17. [17]
    F. Fricker, Einführung in die Gitterpunktlehre, Birkhäuser, Basel, 1982.zbMATHGoogle Scholar
  18. [18]
    W. Gasarch, J. Glenn and C. P. Kruskal, Finding large 3-free sets I: The small n case, Journal of Computer and System Sciences 74 (2008), 628–655.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    B. Green and T. Tao, New bounds for Szemerédi’s theorem, II: A new bound for r4(n), in Analytic Number Theory: essays in honour of Klaus Roth, (W. W. L. Chen, W. T. Gowers, H. Halber Stam, W. M. Schmidt, R. C. Vaughan eds.) 2009, pp. 180–204.Google Scholar
  20. [20]
    B. Green and J. Wolf, A note on Elkin’s improvement of Behrend’s construction, Additive Number Theory (2010), 141–144.Google Scholar
  21. [21]
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th Edition, Oxford Science Publications, Oxford University Press, Oxford, 2004.Google Scholar
  22. [22]
    V. Jarnik, Über Gitterpunkte und konvex Kurven, Mathematische Zeitschrift 2 (1925), 500–518.MathSciNetGoogle Scholar
  23. [23]
    A. Khintchine, A qualitative formulation of Kronecker’s theory of approximation, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 12 (1948), 113–122.MathSciNetGoogle Scholar
  24. [24]
    I. Laba and M. T. Lacey, On sets of integers not containing long arithmetic progressions, ArXiv Mathematics e-prints (2001).Google Scholar
  25. [25]
    L. Moser, On non-averaging sets of integers, Canadian Journal of Mathematics 5 (1953), 245–253.zbMATHCrossRefGoogle Scholar
  26. [26]
    K. O’Bryant, Sets of integers that do not contain long arithmetic progressions, arXiv:0811.3057v2, 2008.Google Scholar
  27. [27]
    R. A. Rankin, Sets not containing more than a given number of terms in arithmetic progression, Proceedings of the Royal Society of Edinburgh. Section A 65 (1960), 332–344.MathSciNetGoogle Scholar
  28. [28]
    K. F. Roth, On certain sets of integers, Journal of the London Mathematical Society 28 (1953), 245–252.CrossRefGoogle Scholar
  29. [29]
    I. Z. Ruzsa, Solving a linear equation on a set of integers I, Acta Arithmetica 65 (1993), 259–282.MathSciNetzbMATHGoogle Scholar
  30. [30]
    R. Salem and D. Spencer, On sets of integers which contain no three in arithmetic progression, Proceedings of the National Academy of Sciences of the United States of America 28 (1942), 561–563.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    A. Shapira, Behrend-type constructions for sets of linear equations, Acta Arithmetica 122 (2006), 17–33.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    C. L. Siegel, Über die Klassenzahl quadratischer Zahlkörper, Acta Arithmetica 1 (1935), 83–86.zbMATHGoogle Scholar
  33. [33]
    E. Szemerédi, On sets of integers containing no κ elements in arithmetic progression, Acta Arithmetica 27 (1975), 299–345.Google Scholar
  34. [34]
    B. L. van der Waerden, Beweis einer Baudetischen Vermutung, Nieuw Archief voor Wiskunde 2(15) (1927), 212–216.Google Scholar

Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael

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