A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over F 2 . We introduce a new technique to prove such correlation bounds with F 2 polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In fact, we prove a more general XOR lemma that extends to arbitrary resilient functions. We conjecture that the technique generalizes to higher degree polynomials as well.

A key ingredient in our new approach is a structural result about the Fourier spectrum of low degree polynomials over F 2 . We show that for any n-variate polynomial p over F 2 of degree at most d , there is a small set S [ n ] of variables, such that almost all of the Fourier mass of p lies on Fourier coefficients that intersect with S . In fact our result is more general, and finds such a set S for any low-dimensional subspace of polynomials. This generality is crucial in deriving the new XOR lemmas.