Theory of Computing
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Title : Dimension-Free $L_2$ Maximal Inequality for Spherical Means in the Hypercube
Authors : Aram W. Harrow, Alexandra Kolla, and Leonard J. Schulman
Volume : 10
Number : 3
Pages : 55-75
URL : https://theoryofcomputing.org/articles/v010a003
Abstract
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We establish the maximal inequality claimed in the title. In
combinatorial terms this has the implication that for sufficiently
small $\epsilon>0$, for all $n$, any marking of an $\epsilon$
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 $x$.