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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.

Sources for this record

What was already known

The slice-density conjecture was known in low dimensions. Lepsveridze and Sun had developed the one-dimensional-fibre and dual-program approach, but not the all-dimensional discrete coupling.

AI involvement AI-centered

Keevash and Lim credit ChatGPT-5.4 with the main ideas of the joint-mixability proof. They formulated the density argument and explain how the coupling supports the all-dimensional result.

The paper explicitly attributes the main ideas of the all-d joint-mixability theorem to ChatGPT-5.4 while the human authors complete the paper and density implication.

Importance Notable

Extends the slice-density result to every dimension.

Limitations

Numerical checks cover an auxiliary slice estimate for 3≤d≤200; they do not check the all-dimensional coupling or full density argument.

Technical verification Partial

The authors report numerical verification of supporting slice-estimate Lemma 3.1 for 3≤d≤200, but no completed rigorous human or machine check of the all-d joint-mixability theorem or full density argument. Numerical tests support only a finite auxiliary range and do not count as a machine checking method.

Human understanding Full

The human authors explain the first-coordinate stationary coupling, induction and dual-program bridge from joint mixability to density.

Correction and publication history