Sharper bounds on the Fourier concentration of DNFs

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 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. The full version of this paper is available at https://arxiv.org/abs/2109.04525. We strongly recommend reading the full version because it has better typesetting.