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A structure theorem for almost low-degree functions on the slice
The Fourier-Walsh expansion of a Boolean function is its unique representation as a multilinear polynomial.
The Kindler-Safra theorem (2002) asserts that if in the expansion of , the
total weight on coefficients beyond degree is very small, then can be
approximated by a Boolean-valued function depending on at most
variables.
In this paper we prove a similar theorem for Boolean functions whose domain
is the `slice' , where , with respect to their unique representation as
harmonic multilinear polynomials. We show that if in the representation of
, the total weight beyond
degree is at most , where , then
can be -approximated by a degree- Boolean function on the
slice, which in turn depends on 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 to
, which is tight in terms of the
dependence on and misses at most a factor of in the
lower-order term.Comment: 30 page