A recent note shows that any subset 𝔽 of the n‑dimensional hypercube {0,1}^n that contains one‑eighth of all points must, on average, be far apart: the average pairwise Hamming distance of points in 𝔽 is at least (n‑3)/2.

The authors prove the lower bound by analyzing the indicator function f of the subset with Fourier analysis. They bound the level‑1 Fourier weight W₁[f] and use inequalities for cosine sums to show that W₁[f] cannot exceed 3/64 for the given density. If equality is reached, the structure of f forces 𝔽 to be an (n‑3)-dimensional subcube.

Equality is achieved only by subcubes of dimension n‑3, confirming a question posed by Lei Yu and resolving a special case of an open problem first stated by Ahlswede and Katona in 1977.

This result settles the minimum average Hamming distance for the particular density 1/8, improving our understanding of distances in Boolean spaces and providing a sharp characterization of extremal subsets.