Dimension-Free $L_2$ Maximal Inequality for Spherical Means in the Hypercube
Received: February 21, 2013
Revised: November 12, 2013
Published: May 23, 2014
Revised: November 12, 2013
Published: May 23, 2014
Keywords: maximal inequality, Fourier analysis, Boolean hypercube
Categories: complexity theory, Boolean functions, Fourier analysis, means, spherical means, Boolean cube, inequality, special issue, Boolean special issue
ACM Classification: G.3
AMS Classification: 42B25
Abstract: [Plain Text Version]
$
\newcommand{\ep}{\varepsilon}
$
We establish the maximal inequality claimed in the title. In combinatorial terms this has the implication that for sufficiently small $\ep>0$, for all $n$, any marking of an $\ep$ fraction of the vertices of the $n$-dimensional hypercube necessarily leaves a vertex $x$ such that marked vertices are a minority of every sphere centered at $x$.
Licensed under a Creative Commons Attribution License (CC-BY)