Dimension-free L2 maximal inequality for spherical means in the   hypercube
  We establish the result of the title.   In combinatorial terms this has the implication that for sufficiently small eps > 0, for all n, any marking of an eps 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. 
Aram W. Harrow, Alexandra Kolla, Leonard J. Schulman
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