Friedgut’s √n-iteration conjecture on intersections of iterated upper shadows

Establish that for any real parameters ζ > 0 and ε > 0, there exists δ > 0 such that the following holds: for every even integer n and every family A of n/2-element subsets of [n] with ζ ≤ μ(A) ≤ 1 − ζ (μ denoting the uniform measure on the middle layer [n choose n/2]), if r = ⌈ε√n⌉ then the r-th iterated upper shadows ∂⁺ʳ(A) and ∂⁺ʳ(A^c) intersect in a set of measure at least δ, i.e., μ(∂⁺ʳ(A) ∩ ∂⁺ʳ(A^c)) ≥ δ.

Background

The paper studies families A of n/2-subsets of [n] and their iterated upper shadows within the discrete cube. For such families, the authors consider the measure μ on the middle layer and analyze intersections between iterated upper shadows of A and its complement A^c.

Friedgut conjectured that iterating the upper shadow only √n times (scaled by ε) suffices to guarantee a positive-measure intersection between ∂⁺ʳ(A) and ∂⁺ʳ(A^c), uniformly over all A with measure bounded away from 0 and 1. This conjecture is stated explicitly as Conjecture 1.