The Largest Sum-Free Subset of the Lattice Cube
Statement
How dense can a sum-free subset of the lattice cube 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 for a linear map , previously known only for . Proved for all . The paper also shows the same phenomenon fails if the cube is replaced by an arbitrary convex set avoiding the origin.
Record
Comments
No person has examined this. Nothing here has been checked at all. say whether it holds →
proof attempt · #1
Peter Keevash and Jeck Lim, using ChatGPT 5.4That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
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.
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