On the ℓ4:ℓ2 ratio of functions with restricted Fourier support

Naomi Kirshner, Alex Samorodnitsky

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Abstract

Given a subset A⊆{0,1}n, let μ(A) be the maximal ratio between ℓ4 and ℓ2 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 languageEnglish
Article number105202
JournalJournal of Combinatorial Theory. Series A
Volume172
DOIs
StatePublished - May 2020
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2020 Elsevier Inc.

Funding

Research partially supported by ISF grant 1724/15.

FundersFunder number
Israel Science Foundation1724/15

    Keywords

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

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