## Abstract

Given a subset A⊆{0,1}^{n}, let μ(A) be the maximal ratio between ℓ_{4} and ℓ_{2} norms of a function whose Fourier support is a subset of A.^{1} We make some simple observations about the connections between μ(A) and the additive properties of A on one hand, and between μ(A) and the uncertainty principle for A on the other hand. One application obtained by combining these observations with results in additive number theory is a stability result for the uncertainty principle on the discrete cube. Our more technical contribution is determining μ(A) rather precisely, when A is a Hamming sphere S(n,k) for all 0≤k≤n.

Original language | English |
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Article number | 105202 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 172 |

DOIs | |

State | Published - May 2020 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2020 Elsevier Inc.

### Funding

Research partially supported by ISF grant 1724/15.

Funders | Funder number |
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Israel Science Foundation | 1724/15 |

## Keywords

- Additive combinatorics
- Fourier spectrum
- Hypercontractivity
- Krawchouk polynomials
- Uncertainty principle

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