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.Comment: 36 page
