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

Combinatorica volume 36, pages229–248(2016)Cite this article

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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Affiliations

  1. CMLS, École polytechnique, 91128, Palaiseau Cedex, France

    Alain Plagne

  2. Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, Vandoeuvre-lès-Nancy, F-54506, France

    Anne de Roton

  3. Institut Elie Cartan de Lorraine, UMR 7502, CNRS, Vandoeuvre-lès-Nancy, F-54506, France

    Anne de Roton

Authors
  1. Alain Plagne
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  2. Anne de Roton
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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