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

Naomi Kirshner; Alex Samorodnitsky

doi:https://doi.org/10.1016/j.jcta.2019.105202

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

Naomi Kirshner, Alex Samorodnitsky

The Rachel and Selim Benin School of Engineering and Computer Science

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

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 language	English
Article number	105202
Journal	Journal of Combinatorial Theory - Series A
Volume	172
DOIs	https://doi.org/10.1016/j.jcta.2019.105202
State	Published - May 2020

Keywords

Additive combinatorics
Fourier spectrum
Hypercontractivity
Krawchouk polynomials
Uncertainty principle

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Access to Document

https://doi.org/10.1016/j.jcta.2019.105202

Cite this

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title = "On the ℓ4:ℓ2 ratio of functions with restricted Fourier support",

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.",

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N2 - 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.

AB - 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.

KW - Additive combinatorics

KW - Fourier spectrum

KW - Hypercontractivity

KW - Krawchouk polynomials

KW - Uncertainty principle

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