A remark on Bourgain’s distributional inequality on the Fourier spectrum of Boolean functions

Abstract

Bourgain’s theorem says that under certain conditions a function $f:\{0,1{\}}^n\to \{0,1\}$ can be approximated by a function $g$ which depends only on a small number of variables. By following his proof we obtain a generalization for the case that there is a nonuniform product measure on the domain of $f$.