Maximum $k$-sum $\mathbf{n}$-free sets of the 2-dimensional integer lattice

For a positive integer $n$, let $[n]$ denote $\{1,\dots ,n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$and$n, a \textit{k$-sum$$\mathbf{b}$-free set} of[n]\times [n]$ is a subset $S$ of $[n]\times [n]$ such that there are no elements ${$\mathbf{a}$}_1, \ldots, {$\mathbf{a}$}_k$in$S$satisfying${$\mathbf{a}$}_1+\cdots+{$\mathbf{a}$}_k =$\mathbf{b}$$\mathrm{.}Fora2-dimensionalintegerlatticepoint$$\mathbf{b}$$ and positive integers $k\ge 2$ and $n$, we determine the maximum density of a {$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions.