Stability of near-extremal sum-free subsets in dimensions at least three

Establish a stability theorem for sum-free subsets of the lattice cube [n]^d for every fixed dimension d≥3, showing that any sum-free set whose size is close to the maximum possible size is close, in symmetric difference, to the optimal stripe.

Background

The paper studies the number of sum-free subsets of the d-dimensional lattice cube [n]^d and relies on the asymptotic determination of the maximum size M([n]^d), achieved up to a boundary-order error by an appropriate stripe. Stability results describe the structure of sets whose size is close to this maximum.

A stability theorem is known in dimension d=1, and for d=2 it is known that near-extremal sum-free subsets are close to the optimal stripe. The paper explicitly identifies the analogous stability problem for dimensions d≥3 as unresolved. In the concluding remarks, the paper indicates that a possible approach would be to trace near-equality cases in the Keevash–Lim dual-weight argument and convert averaged near-equality into structural information on most fibers.