A structure theorem for almost low-degree functions on the slice

Abstract: The Fourier-Walsh expansion of a Boolean function $f \colon {0,1}ⁿ \rightarrow {0,1}$ is its unique representation as a multilinear polynomial. The Kindler-Safra theorem (2002) asserts that if in the expansion of $f$, the total weight on coefficients beyond degree $k$ is very small, then $f$ can be approximated by a Boolean-valued function depending on at most $O(2<sup>k)$ variables. In this paper we prove a similar theorem for Boolean functions whose domain is the `slice' $${{[n]}\choose{pn}} = {x \in {0,1}^n\colon \sum_i x_i = pn}$$, where $0 \ll p \ll 1$, with respect to their unique representation as harmonic multilinear polynomials. We show that if in the representation of $f\colon {{[n]}\choose{pn}} \rightarrow {0,1}$, the total weight beyond degree $k$ is at most $\epsilon$, where $\epsilon = \min(p, 1-p)<sup>{O(k)}$, then $f$ can be $O(\epsilon)$-approximated by a degree-$k$ Boolean function on the slice, which in turn depends on $O(2<sup>{k})$ coordinates. This proves a conjecture of Filmus, Kindler, Mossel, and Wimmer (2015). Our proof relies on hypercontractivity, along with a novel kind of a shifting procedure. In addition, we show that the approximation rate in the Kindler-Safra theorem can be improved from $\epsilon + \exp(O(k)) \epsilon<sup>{1/4}$ to $$\epsilon+\epsilon² (2\ln(1/\epsilon))<sup>k/k!$$, which is tight in terms of the dependence on $\epsilon$ and misses at most a factor of $2<sup>{O(k)}$ in the lower-order term.