On $\ell_4 : \ell_2$ ratio of functions with restricted Fourier support

TL;DR

This paper investigates the ratio of $ ext{ell}_4$ to $ ext{ell}_2$ norms for functions with Fourier support in subsets of the hypercube, linking it to additive properties and the uncertainty principle, with precise results for Hamming spheres.

Contribution

It provides new insights into the $ ext{ell}_4$ to $ ext{ell}_2$ ratio for functions supported on Hamming spheres and explores connections with additive combinatorics and the uncertainty principle.

Findings

Determined $ ext{ell}_4$ to $ ext{ell}_2$ ratio for functions supported on Hamming spheres.

Established connections between this ratio, additive properties, and the uncertainty principle.

Provided a stability result for the uncertainty principle on the discrete cube.

Abstract

Given a subset $A \subseteq \{0,1\}^n$, let $\mu(A)$ be the maximal ratio between $\ell_4$ and $\ell_2$ norms of a function whose Fourier support is a subset of $A$. We make some simple observations about the connections between $\mu(A)$ and the additive properties of $A$ on one hand, and between $\mu(A)$ 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 $\mu(A)$ rather precisely, when $A$ is a Hamming sphere $S(n,k)$ for all $0 \le k \le n$.