Bounded Relative Boundary Implies Narrow DNF Approximation

2026-08-31

AI Summary

本文证明了Friedgut关于有界相对边界的递增家族可被有限大小元素的集合良好近似的猜想，通过构建特定宽度的单调DNF来实现。

Abstract

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.

Problem

• Bounded Relative Boundary
• Increasing Family
• Approximation
• DNF
• Discrete Cube

Innovation

• Bounded Relative Boundary
• Narrow DNF Approximation
• Randomized Shifting Procedure