Comptes Rendus
Article de recherche - Combinatoire, Analyse harmonique
Spherical sets avoiding orthonormal bases
[Ensembles sphériques évitant des bases orthonormées]
Dmitrii Zakharov1
1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1351-1362

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

Nous montrons qu’il existe une constante absolue $c_0<1$ telle que pour tout $n \geqslant 2$, tout ensemble mesurable $A \subset S^{n-1}$ de densité au moins $c_0$ contient $n$ vecteurs orthogonaux deux à deux. Le résultat est optimal à la valeur de la constante $c_0$ près. De plus, nous montrons que pour tout $2 \leqslant k \leqslant n$ un ensemble $A$ évitant $k$ vecteurs orthogonaux deux à deux a une mesure au plus égale à $\exp \bigl (-c_1 \min \bigl \lbrace \sqrt{n}, n/k\bigr \rbrace \bigr )$ pour $c_1>0$. Les démonstrations reposent sur l’analyse harmonique sur la sphère et l’inégalité d’hypercontractivité.

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DOI : 10.5802/crmath.782
Classification : 52C10, 33C45

Dmitrii Zakharov 1

1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Spherical sets avoiding orthonormal bases},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1351--1362},
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     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.782},
     language = {en},
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Dmitrii Zakharov. Spherical sets avoiding orthonormal bases. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1351-1362. doi: 10.5802/crmath.782

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