Dimension-free L2 maximal inequality for spherical means in the hypercube

Abstract

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.

17 pages. v2. This version matches the published version and fixes typos, simplifies some proofs and makes other minor changes. The main results are unchanged from v1