有界相对边界蕴含窄DNF近似

中文摘要 AI 辅助

弗里德古特（Friedgut）猜想，在p偏置离散立方体中具有有界相对边界的递增族，可被其极小元大小有界的族任意逼近，且该界与维度和偏置无关（《美国数学会杂志》12卷，1999年）。我们证明了该猜想：对0<p≤1/2，总重采样影响至多为K的每个递增布尔函数，在μ_p^n下与宽度为exp(O((K+1)²/ε²))的单调DNF是ε-接近的；针对高偏置的独立论证完成了所有p∈(0,1)的证明。我们的证明基于哈塔米（Hatami）的伪 Junta 定理（《数学年刊》176卷，2012年），追踪哈塔米的构造分离出具有递增局部激活、无维度元数及多重计数负载界的自适应表示。我们的主要新要素是偏置匹配的随机移位过程，该过程将伪 Junta 逼近器转换为递增函数，同时保留其自适应表示的受控强制细化的精确可测性；从所得单调自适应表示中，我们提取正证书并截断以获得所需的窄DNF。

英文摘要

Friedgut conjectured that an increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have bounded size, with a bound independent of the dimension and the bias (J. Amer. Math. Soc. 12 (1999)). We prove this conjecture by showing that, for $0<p\leq 1/2$, every increasing Boolean function with total resampling influence at most $K$ is $\varepsilon$-close under $μ_p^n$ to a monotone DNF of width $\exp(O((K+1)^2/\varepsilon^2))$. A separate high-bias argument completes the proof for all $p\in(0,1)$. Our proof builds on Hatami's pseudo-junta theorem (Ann. of Math. 176 (2012)). Tracking Hatami's construction isolates an adaptive representation with increasing local activations and dimension-free arity and multiplicity-counted load bounds. Our main new ingredient is a bias-matched randomized shifting procedure that converts the pseudo-junta approximator into an increasing function while retaining exact measurability with respect to a controlled forced refinement of its adaptive representation. From the resulting monotone adaptive representation, we extract positive certificates and truncate them to obtain the required narrow DNF.

发表机构

• Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）
• University of Regensburg（雷根斯堡大学）

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