Dimension-Free $L_2$ Maximal Inequality for Spherical Means in the Hypercube
Revised: November 12, 2013
Published: May 23, 2014
$\newcommand{\ep}{\varepsilon}$
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 $x$.