In this talk, we establish a dimension-free $\ell_2$ maximal inequality
for spherical means on the Hypercube graph $\{0,1\}^n$. We explain
possible connections to the notorious Unique Games Conjecture.

Combinatorially, this inequality implies the following stronger
alternative to the union-bound technique: Assume that we have a binary
function f (values 0,1) on the n-dimensional hypercube $H_n$, with $N=2^n$
vertices. Think of the set $X=\{x \in H_n : f(x)=1\}$ as the set vertices
that a malicious adversary "spoils". Assume ${|X| \le \epsilon N}$, i.e. the
adversary can only spoil up to $\epsilon$ fraction of the vertices. Fix
a threshold ${\lambda \le \epsilon}$.