2026/07/28 by Ting-Wei Chao, Zixuan Xu, Dmitrii Zakharov
#math.CO
Given a subset A ⊆ \mathbb F2n, we can consider the distribution of the intersection size of A with a uniformly random d-flat F. Motivated by the edge statistics problem and the hypercube statistics problem, the affine subspace statistics problem concerns the maximum of ℙ[|F∩ A|=s] among A ⊆ \mathbb F2n for any fixed s∈\1,…,2d\ over a uniformly random d-flat F. We use λ^*(d,s) to denote the limit of the maximum when n goes to infinity. In this note, we prove tight bounds for λ^*(d,s) in two different regimes. For s=j2k where j is a positive odd integer, the best known lower bound construction achieving λ^*(d,s)≥ 1-2-k is due to taking A as the union of j parallel (n-d+k)-flats in \mathbb F2n. Our main result is a matching upper bound with an additive error term of O(2-3k/2). We also study the case s=1, where we determine λ^*(d,1) exactly. We show that the random construction where each point is included with probability 2-d is optimal.