Year: 2025
Type: article
Abstract: We show that there exists an absolute constant c 0 < 1 such that for all n ⩾ 2 , any measurable set A ⊂ 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 ⩽ k ⩽ n a set A avoiding k pairwise orthogonal vectors has measure at most exp - c 1 min n , n / k for some c 1 > 0 . Proofs rely on the harmonic analysis on the sphere and the hypercontractive inequality.
Source: Comptes Rendus Mathématique
DOI: 10.5802/crmath.782
Author: Dmitrii Zakharov
Institution: Massachusetts Institute of Technology
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4415805676