Spherical sets avoiding orthonormal bases

We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any measurable 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 $c_0$. Moreover, we show that for all $2\le k \le n$ a set $A$ avoiding $k$ pairwise orthogonal vectors has measure at most $\exp(-c_1 \min\{\sqrt{n}, n/k\})$ for some $c_1>0$. Proofs rely on the harmonic analysis on the sphere and the hypercontractive inequality.