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

We determine the shape of all sum-free sets in ${1,dots,n}^2$ 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 $frac{4}{5}n-o(n)le x+ylefrac{8}{5}n+ o(n)$. We also determine for any positive integer $p$ the maximum size of a subset $Asubseteq {1,dots,n}^2$ which forbids the triple $(x,y,z)$ satisfying $px+py=z$.