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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 ${\\mathbf{a}}_1+\\cdots+{\\mathbf{a}}_k =\\mathbf{b}$. 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 $[n]\\times [n]$. 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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 ${\\mathbf{a}}_1+\\cdots+{\\mathbf{a}}_k =\\mathbf{b}$. 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 $[n]\\times [n]$. 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