On the largest sum-free subset of the lattice cube

TL;DR

This paper determines the limiting density of the largest sum-free subset within lattice cubes in all dimensions, confirming the conjecture that such sets are formed by two hyperplane slices.

Contribution

It provides a complete solution for the limiting density of largest sum-free subsets in lattice cubes across all dimensions, resolving a natural conjecture.

Findings

The limiting density is explicitly determined for all dimensions.

The largest sum-free subsets are constructed by two hyperplane slices.

The conjecture about the structure of these sets is confirmed.

Abstract

We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices.