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

TL;DR

This paper proves a dimension-free L2 maximal inequality for spherical means in the hypercube, showing that small fractions of marked vertices cannot dominate on all spheres centered at any vertex.

Contribution

It establishes a new dimension-free inequality for spherical means in the hypercube, with implications for combinatorial vertex marking.

Findings

Dimension-free L2 maximal inequality proved

Marked vertices cannot dominate on all spheres for small fractions

Implications for combinatorial vertex marking

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.