The Largest Sum-Free Subset of the Lattice Cube

How dense can a sum-free subset of the lattice cube $\{1,\dots,n\}^d$ be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice $\{x : 1 \le L(x) < 2\}$ for a linear map $L$ , previously known only for $d \le 4$ . Proved for all $d$ . The paper also shows the same phenomenon fails if the cube is replaced by an arbitrary convex set avoiding the origin.

Result: Proved
Status: Resolved
AI contribution: AI co-developed
Method: Argument
Field: Additive combinatorics
Posed by: Harout Aydinian, Peter Cameron; Problem 6 in Ben Green's list of 100 open problems
Year posed: —
Years open: —
Solved: 2026-05-01
Model: ChatGPT 5.4
Vendor: OpenAI
Collaborators: Peter Keevash, Jeck Lim
Verification: Unreviewed
Publication: Preprint
Significance: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

One line in the acknowledgements, scoped to one theorem: ChatGPT-5.4 provided the main ideas used in the proof of Theorem 1.5, and helped generate the code for numerically verifying a lemma at small parameters. That theorem is not incidental. The authors call it the main contribution of the paper: a general joint mixability statement in the discrete setting which implies the coupling conjecture that Lepsveridze and Sun had reduced the problem to, and which is what carries the density result to all dimensions.

Verification

arXiv preprint, not peer-reviewed. One lemma is verified numerically for small parameters in an appendix; the rest is a linear programming duality argument.

Source

arXiv:2605.00816 - On the largest sum-free subset of the lattice cube

Submitted by Curator34

The Largest Sum-Free Subset of the Lattice Cube

What the AI did

Verification

Source

Discussion