A structure theorem for almost low-degree functions on the slice
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The Fourier-Walsh expansion of a Boolean function f {0,1}^n →{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^k) variables. In this paper we prove a similar theorem for Boolean functions whose domain is the `slice' [n]pn = {x ∈{0,1}^n∑_i x_i = pn}, where 0 ≪ p ≪ 1, with respect to their unique representation as harmonic multilinear polynomials. We show that if in the representation of f[n]pn→{0,1}, the total weight beyond degree k is at most ϵ, where ϵ = (p, 1-p)^O(k), then f can be O(ϵ)-approximated by a degree-k Boolean function on the slice, which in turn depends on O(2^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 ϵ + (O(k)) ϵ^1/4 to ϵ+ϵ^2 (2(1/ϵ))^k/k!, which is tight in terms of the dependence on ϵ and misses at most a factor of 2^O(k) in the lower-order term.
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