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

## TL;DR

This paper determines the maximum density of sets in a 2D integer lattice that avoid certain sum configurations, extending sum-free set theory to higher dimensions and non-homogeneous cases.

## Contribution

It introduces the first study of non-homogeneous sum-free sets in higher dimensions, specifically for 2D lattice points and arbitrary sums.

## Key findings

- Established the maximum density of $k$-sum $ extbf{b}$-free sets in $[n]\times [n]$
- Extended sum-free set theory to non-homogeneous, higher-dimensional contexts
- Provided foundational results for future research in multi-dimensional additive combinatorics.

## Abstract

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 \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 $[n]\times [n]$. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions.