Sharper bounds on the Fourier concentration of DNFs

Abstract

In 1992 Mansour proved that every size-s DNF formula is Fourier-concentrated on s O(log log s) coefficients. We improve this to s O(log log k) where k is the read number of the DNF. Since k is always at most s, our bound matches Mansour's for all DNFs and strengthens it for small-read ones. The previous best bound for read-k DNFs was s O(k 3/2 ) . For k up to Θ(log log s), we further improve our bound to the optimal poly(s); previously no such bound was known for any k = ωs(1).Our techniques involve new connections between the term structure of a DNF, viewed as a set system, and its Fourier spectrum.