Invariance principle on the slice

TL;DR

This paper establishes an invariance principle for low-degree, low-influence functions on the slice of the Boolean cube, connecting their behavior across different spaces and deriving several important corollaries.

Contribution

It introduces a new invariance principle for functions on the slice, extending analysis tools to this setting and deriving multiple classical results as corollaries.

Findings

Proves invariance principle for low-influence functions on the slice

Derives a majority is stablest theorem for the slice

Provides a stability version of Wilson's theorem for t-intersecting families

Abstract

We prove an invariance principle for functions on a slice of the Boolean cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our invariance principle shows that a low-degree, low-influence function has similar distributions on the slice, on the entire Boolean cube, and on Gaussian space. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. Our result imply a version of majority is stablest for functions on the slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra theorem. As a corollary of the Kindler-Safra theorem, we prove a stability result of Wilson's theorem for t-intersecting families of sets, improving on a result of Friedgut.