Abstract
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(2k) variables.
In this paper we prove a similar theorem for Boolean functions whose domain is the ‘slice’ \(\left({\matrix{{\left[n \right]} \cr {pn} \cr}} \right) = {\rm{\{}}x \in {{\rm{\{}}0,1{\rm{\}}}^n}:\sum\nolimits_i {{x_i} = pn} {\rm{\}}}\), where 0 ≪ p ≪ 1, with respect to their unique representation as harmonic multilinear polynomials. We show that if in the representation of \(f:\left({\matrix{{\left[n \right]} \cr {pn} \cr}} \right) \to {\rm{\{}}0,1{\rm{\}}}\), the total weight beyond degree k is at most ϵ, where ϵ =min(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(2k) 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 + exp(O(k))ϵ5/4 to ϵ + ϵ2(2 ln(1/ϵ))k/k!, which is tight in terms of the dependence on ϵ and misses at most a factor of 2O(k) in the lower-order term.
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Acknowledgements
We are deeply grateful to Yuval Filmus for numerous helpful comments and suggestions, and wish to thank Guy Kindler for explaining to us some aspects of his work [33]. We are also grateful to the anonymous referee for constructive comments which helped us to improve the presentation.
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Research supported by the Israel Science Foundation (grants no. 402/13 and 1612/17) and the Binational US-Israel Science Foundation (grant no. 2014290).
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Keller, N., Klein, O. A structure theorem for almost low-degree functions on the slice. Isr. J. Math. 240, 179–221 (2020). https://doi.org/10.1007/s11856-020-2062-4
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DOI: https://doi.org/10.1007/s11856-020-2062-4