On $\ell_4 : \ell_2$ ratio of functions with restricted Fourier support
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 $\...
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Zusammenfassung: | 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 $0 \le k \le n$. |
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DOI: | 10.48550/arxiv.1801.08507 |