Definition.Spectral Sample [boolean/def/spec-sample]

For $𝑓:{\left\{\pm 1\right\}}^{𝑛}\to \left\{\pm 1\right\}$, the spectral sample for $𝑓$, notated ${\mathrm{𝒮︀}}_{𝑓}$, is the probability distribution on subsets of $\left[𝑛\right]$ in which $𝐏𝐫\left[𝑆\right]=\widehat{𝑓}{\left(𝑆\right)}^2$.

We write $𝑺\sim {\mathrm{𝒮︀}}_{𝑓}$ for a set drawn from this distribution.