Maximum $k$-Sum $\mathbf{n}$-Free Sets of the 2-Dimensional Integer Lattice

For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a $k$-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 $\mathbf{a}_1+\cdots+\mathbf{a}_k =\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ 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.