← 研踪 PaperTrailarXiv:2609.00240 · v2
Bounded Relative Boundary Implies Narrow DNF Approximation
首次提交 2026-09-01 02:43
版本更新 2026-11-06 22:57
雷达观测 2026-11-09 13:22
摘要
We prove Friedgut's conjecture (J. Amer. Math. Soc. 12 (1999)) that every increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have size bounded independently of the dimension and the bias. For $0<p\le 1/2$, every increasing Boolean function with total resampling influence at most $K$ is $\varepsilon$-close under $μ_p^n$ to a monotone DNF of width $\exp(O((K+1)^2/\varepsilon^2))$, from which the full conjecture follows for all $p\in(0,1)$. We build on Hatami's pseudo-junta theorem (Ann. of Math. 176 (2012)), whose approximator need not be increasing. Our randomized shifts produce an increasing approximator that is measurable with respect to a controlled refinement of Hatami's adaptive partition. Retaining only short positive certificates yields the desired narrow DNF.
作者备注：13 pages; proof and exposition revised, Lean repository linked
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v2 · 2026-11-06 22:57 · 最新观测版本
Bounded Relative Boundary Implies Narrow DNF Approximation
We prove Friedgut's conjecture (J. Amer. Math. Soc. 12 (1999)) that every increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have size bounded independently of the dimension and the bias. For $0<p\le 1/2$, every increasing Boolean function with total resampling influence at most $K$ is $\varepsilon$-close under $μ_p^n$ to a monotone DNF of width $\exp(O((K+1)^2/\varepsilon^2))$, from which the full conjecture follows for all $p\in(0,1)$. We build on Hatami's pseudo-junta theorem (Ann. of Math. 176 (2012)), whose approximator need not be increasing. Our randomized shifts produce an increasing approximator that is measurable with respect to a controlled refinement of Hatami's adaptive partition. Retaining only short positive certificates yields the desired narrow DNF.
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