Spherical sets avoiding orthogonal bases

We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$ pairwise orthogonal vectors. The result is sharp up to the value of the constant $c_0$. The proof relies on the harmonic analysis on the sphere and hypercontractivity in the Gaussian space.