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Revision #1 to TR18-016 | 10th June 2018 13:45
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#### On $\ell_4$ : $\ell_2$ ratio of functions with restricted Fourier support

**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 $0\leq k\leq n$.

**Changes to previous version:**
In this revision, we simplify the proof of Proposition 4.5, add a reference to a paper of Polyanskiy, fix a typo in the first author's name, replace a wrong argument on page 13, fifth line from the bottom, and add acknowledgements.

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TR18-016 | 25th January 2018 18:23
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#### On $\ell_4$ : $\ell_2$ ratio of functions with restricted Fourier support

**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 $0\leq k\leq n$.