Maximum sum-free density in lattice cubes

For lattice-cube slices L_d(r), L_d(s), L_d(t) with nonnegative r+s+t=dn, ask whether their uniform distributions are jointly mixable in every d≥1. The application targets the optimal limiting sum-free density of [n+1]^d with O(n^(d−1)) error.

Why it matters and what counts as progress

Why it matters: It connects coupling of slice distributions to sharp additive extremal density in every dimension.

What counts as progress: Prove joint mixability of all the stated cube slices and obtain the sharp sum-free density with controlled boundary error.

How the problem developed

1 May 2026 · Proof [1]
Peter Keevash and Jeck Lim Determined the limiting maximum density of sum-free subsets of [n]ᵈ as n grows, for every dimension d.

AI contribution

On the largest sum-free subset of the lattice cube

Peter Keevash, Jeck Lim

For every d≥1 and nonnegative r,s,t with r+s+t=dn, the uniform distributions on the lattice-cube slices L_d(r), L_d(s), and L_d(t) are jointly mixable. Keevash and Lim use this theorem to obtain the optimal limiting sum-free density of [n+1]^d and an O(n^(d−1)) upper error.

AI involvementAI-centered

ImportanceNotable

Technical verificationPartial

Human understandingFull