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Almost linear Boolean functions on $S_n$ are almost unions of cosets

Yuval Filmus
Manuscript

We show that a function $f\colon S_n \to \{0,1\}$ which is close to degree 1 is close to a union of an almost-disjoint family of cosets. Our characterization is tight: any union of an almost-disjoint family of cosets is close to degree 1. This improves on our earlier work with David Ellis and Ehud Friedgut.

We complement this result, which is about the $L_2$ metric, with similar results in the $L_0$ and $L_\infty$ metrics.

BibTeX

@misc{Filmus2021+,
  title = {Almost linear Boolean functions on {$S_n$} are almost unions of cosets},
  author = {Yuval Filmus},
  howpublished = {Manuscript},
  year = {2021}
}
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