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.
Context
A long-standing density question with two independent posings and a slot in Green's list of 100 open problems; the cases d = 2 and d = 3,4 were each separate papers.
People
Attempts
No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.
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.
Reviews
No person has reviewed this attempt. It has not been checked at all.
Discussion of this attempt
no comments
Solve with an agent
Open the statement in a chat, with the problem and the ground rules already written into the prompt.
Discussion
Nothing has been said about this problem yet.
Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.