## 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(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:∑ixi=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 =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(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 + exp(O(k))ϵ^{5/4} to ϵ + ϵ^{2}(2 ln(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.

Original language | English |
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Pages (from-to) | 179-221 |

Number of pages | 43 |

Journal | Israel Journal of Mathematics |

Volume | 240 |

Issue number | 1 |

DOIs | |

State | Published - Oct 2020 |

### Bibliographical note

Publisher Copyright:© 2020, The Hebrew University of Jerusalem.

### Funding

Research supported by the Israel Science Foundation (grants no. 402/13 and 1612/17) and the Binational US-Israel Science Foundation (grant no. 2014290). Acknowledgements

Funders | Funder number |
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Israel Science Foundation | 2014290, 402/13, 1612/17 |