ECCC-Report TR18-016https://eccc.weizmann.ac.il/report/2018/016Comments and Revisions published for TR18-016en-usSun, 10 Jun 2018 13:45:10 +0300
Revision 1
| On $\ell_4$ : $\ell_2$ ratio of functions with restricted Fourier support |
Naomi Kirshner,
Alex Samorodnitsky
https://eccc.weizmann.ac.il/report/2018/016#revision1Given 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 $0\leq k\leq n$.Sun, 10 Jun 2018 13:45:10 +0300https://eccc.weizmann.ac.il/report/2018/016#revision1
Paper TR18-016
| On $\ell_4$ : $\ell_2$ ratio of functions with restricted Fourier support |
Naomi Kirshner,
Alex Samorodnitsky
https://eccc.weizmann.ac.il/report/2018/016Given 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 $0\leq k\leq n$.Fri, 26 Jan 2018 16:00:46 +0200https://eccc.weizmann.ac.il/report/2018/016