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.