On solution-free sets of integers II

Robert Hancock¹ ; Andrew Treglown¹

¹ School of Mathematics University of Birmingham Edgbaston, Birmingham, B15 2TT, UK

Acta Arithmetica, Tome 180 (2017) no. 1, pp. 15-33

Résumé

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.

Détail

DOI : 10.4064/aa8522-6-2017

Keywords: given linear equation mathcal set subseteq mathcal free does contain non trivial solutions mathcal determine precise size largest mathcal free subset several general classes linear equations mathcal form fixed mathbb where geq geq further linear equations mathcal upper bound number maximal mathcal free subsets geq bound exact error term exponent make container removal lemmas green prove result results extend various linear equations three variables