Spherical sets avoiding orthonormal bases

TL;DR

This paper proves that large enough measurable sets on the sphere necessarily contain orthogonal vectors, and characterizes the measure of sets avoiding multiple orthogonal vectors using harmonic analysis and hypercontractivity.

Contribution

It establishes a sharp threshold for the measure of sets containing orthogonal vectors and provides bounds for sets avoiding multiple orthogonal vectors.

Findings

Sets with density above a constant contain orthogonal vectors

Measure bounds for sets avoiding k orthogonal vectors

Use of harmonic analysis and hypercontractivity in proofs

Abstract

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.