Maximal sets with no solution to x+y=3z

Abstract

In this paper, we are interested in a generalization of the notion of sum-free sets. We address a conjecture rst made in the 90s by Chung and Goldwasser. Recently, after some computer checks, this conjecture was formulated again by Matolcsi and Ruzsa, who made a rst signicant step towards it. Here, we prove the full conjecture by giving an optimal upper bound for the Lebesgue measure of a 3-sum-free subset A of [0; 1], that is, a set containing no solution to the equation x+y=3z where x, y and z are restricted to belong to A. We then address the inverse problem and characterize precisely, among all sets with that property, those attaining the maximal possible measure.

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References

  1. [1]

    A. Baltz, P. Hegarty, J. Knape, U. Larsson and T. Schoen: The structure of maximum subsets of {1,…, n} with no solutions to a+b=kc, Electron. J. Combin. 12 (2005), Research Paper 19.

  2. [2]

    P. Candela and O. Sisask: On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime, Acta Math. Hungar. 132 (2011), 223–243.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    F. R. K. Chung and J. L. Goldwasser: Maximum subsets of [0; 1] with no solutions to x+y=kz, Electron. J. Combin. 3 (1996).

  4. [4]

    F. R. K. Chung and J. L. Goldwasser: Integer Sets Containing no Solution to x+y=3z, The mathematics of Paul Erdős (1997), Springer, 218-227.

  5. [5]

    J.-M. Deshouillers, G. A. Freiman, V. Sós and M. Temkin: On the structure of sum-free sets, 2, Astérisque 258 (1999), 149–161.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    K. Dilcher and L. Lucht: Onnite pattern-free sets of integers, Acta Arith. 121 (2006), 313–325.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    R. Henstock and A. M. Macbeath: On the measure of sum sets, I. The theorems of Brunn, Minkowski and Lusternik, Proc. London Math. Soc. 3 (1953), 182–194.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    L. Lucht: Dichteschranken für die Lösbarkeit gewisser linearer Gleichungen, J. Reine Angew. Math. 285 (1976), 209–217.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    M. Matolcsi and I. Z. Ruzsa: Sets with no solutions to x+y=3z, Europ. J. Combin. 34 (2013), 1411–1414.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    K. F. Roth: On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    I. Z. Ruzsa: Diameter of sets and measure of sumsets, Monatsh. Math. 112 (1991),323–328.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    I. Z. Ruzsa: Solving a linear equation in a set of integers I, Acta Arith. 65 (1993), 259–282.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    I. Z. Ruzsa: Solving a linear equation in a set of integers II, Acta Arith. 72 (1995), 385–397.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    T. Sanders: On Roth’s theorem on progressions, Ann. of Math. 174 (2011), 619–636.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    I. Schur: Über die Kongruenz x m +y m = z m (mod p), Jahresber. Deutsch. Math.-Verein. 25 (1917), 114–117.

    Google Scholar 

  16. [16]

    W. Sierpiński: Surla question de la mesurabilité de la base de M. Hamel, Fund. Math 1 (1920), 105–111.

    MATH  Google Scholar 

  17. [17]

    W. D. Wallis, A. P. Street and J. S. Wallis: Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Mathematics 292, Springer, 1972.

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Correspondence to Alain Plagne.

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Both authors are supported by the ANR grant Cæsar, number ANR 12 - BS01 - 0011.

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Plagne, A., de Roton, A. Maximal sets with no solution to x+y=3z . Combinatorica 36, 229–248 (2016). https://doi.org/10.1007/s00493-015-3100-4

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Mathematics Subject Classication (2000)

  • 05D05
  • 11P99