Maximum \(k\)-sum \(\mathbf{n}\)-free sets of the 2-dimensional integer lattice (Q2205110)

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 \(n\), the authors determined 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.