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In 1992 Mansour proved that every size-$s$ DNF formula isFourier-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 $k$ isalways at most $s$, our bound matches Mansour's for all DNFs and strengthens itfor 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 furtherimprove our bound to the optimal $\mathrm{poly}(s)$; previously no such boundwas 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.

Sharper bounds on the Fourier concentration of DNFs