Sharper bounds on the Fourier concentration of DNFs

TL;DR

This paper improves bounds on the Fourier concentration of DNFs, especially for small read numbers, providing tighter and sometimes optimal bounds, and introduces new techniques linking DNF structure to Fourier spectrum.

Contribution

The authors establish sharper bounds on Fourier concentration for DNFs based on read number, improving previous results and introducing novel analytical techniques.

Findings

Bound of $s^{O( ext{log log }k)}$ for Fourier concentration of size-$s$ DNFs.

Improved bounds for read-$k$ DNFs up to polynomial in $s$ for small $k$.

New connections between DNF term structure and Fourier spectrum.

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 $\tilde{\Theta}(\log\log s)$, we further improve our bound to the optimal $\mathrm{poly}(s)$; previously no such bound was known for any $k = \omega_s(1)$. Our techniques involve new connections between the term structure of a DNF, viewed as a set system, and its Fourier spectrum.