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On solution-free sets of integers II

Volume 180 / 2017

Robert Hancock, Andrew Treglown Acta Arithmetica 180 (2017), 15-33 MSC: Primary 11B75, 05C69. DOI: 10.4064/aa8522-6-2017 Published online: 1 August 2017

Abstract

Given a linear equation $\mathcal{L}$, a set $A \subseteq [n]$ is $\mathcal{L}$-free if $A$ does not contain any ‘non-trivial’ solutions to $\mathcal{L}$. We determine the precise size of the largest $\mathcal{L}$-free subset of $[n]$ for several general classes of linear equations $\mathcal{L}$ of the form $px+qy=rz$ for fixed $p,q,r \in \mathbb N$ where $p \geq q \geq r$. Further, for all such linear equations $\mathcal L$, we give an upper bound on the number of maximal $\mathcal{L}$-free subsets of $[n]$. When $p=q\geq 2$ and $r=1$ this bound is exact up to an error term in the exponent. We make use of container and removal lemmas of Green to prove this result. Our results also extend to various linear equations with more than three variables.

Authors

  • Robert HancockSchool of Mathematics
    University of Birmingham
    Edgbaston, Birmingham, B15 2TT, UK
    e-mail
  • Andrew TreglownSchool of Mathematics
    University of Birmingham
    Edgbaston, Birmingham, B15 2TT, UK
    e-mail

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