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arXiv · 2602.14623

Kakeya conjecture and High-Rank Lattice von Neumann algebras

Mikael de la Salle
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Abstract

If the non-commutative L p space of SLn(Z) has the completely bounded approximation property for some non-trivial value of p, then some form of the Kakeya conjecture holds in dimension d, for all d $\le$ n+1 2 . The proof relies on a spherical analogue of the following question in Euclidean harmonic analysis, that we raise and investigate: does a radially symmetric Fourier multiplier that is bounded on Lp(R d ) for some p __ = 2 necessarily have a continuous symbol? We leave the question open, but we prove that the primitive of such function is smooth in the sense of Zygmund, give some necessary conditions for Lp-boundedness in terms of Besov spaces and Littlewood-Paley decomposition for the symbol, and observe that a negative answer implies some form of the Kakeya conjecture in dimension d. We then provide spherical forms of these results, which, when combined with a refinement of Lafforgue's rank 0 reduction, leads to the claimed result.

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Mikael de la Salle. 2026-02-16. Kakeya conjecture and High-Rank Lattice von Neumann algebras. https://arxiv.org/abs/2602.14623

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