The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice.As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.

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多层的 FKN 定理及其应用

Friedgut-Kalai-Naor (FKN) 定理指出，如果 ƒ 是布尔立方体上接近一阶的布尔函数，则 ƒ 接近于独裁者，一个依赖于单个坐标的函数。作者将定理推广到片，由所有具有固定汉明权重的向量组成的布尔立方体的子集。我们将定理进一步推广到多层，切片的多色版本。作为应用程序，我们证明了边缘等周不等式的稳定性版本，用于参数设置，其中最优集是独裁者。