Shape of the asymptotic maximum sum-free sets in integer lattice grids

Abstract

We determine the shape of all sum-free sets in $\{1,2,\dots ,n{\}}^2$ of size close to the maximum $\frac{3}{5}{n}^2$, solving a problem of Elsholtz and Rackham. We show that all such asymptotic maximum sum-free sets lie completely in the stripe $\frac{4}{5}n-o(n)\le x+y\le \frac{8}{5}n+o(n)$. We also determine for any positive integer $p$ the maximum size of a subset $A\subseteq \{1,2,\dots ,n{\}}^2$ which forbids the triple $(x,y,z)$ satisfying $px+py=z$.

18 pages, 8 figures

References in corpus (1)

• On the structure of large sum-free sets of integers