Intersections of iterated shadows

TL;DR

This paper proves that for large subsets of the middle layer of the Boolean cube with measures bounded away from zero and one, their iterated shadows intersect significantly, confirming a conjecture of Friedgut and relating to the Kruskal--Katona theorem.

Contribution

It establishes a strong form of Friedgut's conjecture by showing intersection properties of iterated shadows for large subsets, providing a stability version of the Kruskal--Katona theorem.

Findings

Iterated shadows of large subsets intersect in positive measure.

Confirms Friedgut's conjecture in a strong form.

Provides a stability result for the Kruskal--Katona theorem.

Abstract

We show that if $\mathcal{A} \subset {[n] \choose n/2}$ with measure bounded away from zero and from one, then the $\Omega(\sqrt{n})$-iterated upper shadows of $\mathcal{A}$ and $\mathcal{A}^c$ intersect in a set of positive measure. This confirms (in a strong form) a conjecture of Friedgut. It can be seen as a stability result for the Kruskal--Katona theorem.