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.

中文翻译：

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DNF 的傅立叶浓度的更清晰界限

1992 年 Mansour 证明了每个 size-$s$ DNF 公式都集中在 $s^{O(\log\log s)}$ 系数上。我们将其改进为 $s^{O(\log\log k)}$，其中 $k$ 是 DNF 的读取数。由于 $k$ 始终最多为 $s$，因此我们的边界匹配所有 DNF 的 Mansour 边界，并针对小读数增强它。之前 read-$k$ DNF 的最佳边界是 $s^{O(k^{3/2})}$。对于高达 $\tilde{\Theta}(\log\log s)$ 的 $k$，我们进一步改进了对最优 $\mathrm{poly}(s)$ 的界限；以前对于任何 $k = \omega_s(1)$ 都不知道这样的界限。我们的技术涉及 DNF 的期限结构（被视为集合系统）与其傅立叶频谱之间的新联系。