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.

翻译：1992年,曼苏尔证明,每个大小-美元DNF公式都是Fleier-$O(\log\logs s)美元系数集中的Fleier。我们把它改进为$O(\log\log k)美元,因为美元是DNF的读数。由于美元总是以美元为单位,因此我们所有DNF的Mansour公式都与Mansour的公式相匹配,小NF的公式得到了加强。上一个读-k美元DNF的公式是$@O(kç3/2})美元。对于美元,最多为$\tilde_theta}(\log\log\logs s)美元,我们进一步改进了我们与美元的最佳值$\mathrm{poly}(s)美元的界限;以前,任何美元=\omega_s(1)美元都没有这种界限。我们的技术涉及DNF的术语结构(被视为一个固定系统)与其四光谱之间的新联系。