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Kakeya conjecture and High-Rank Lattice von Neumann algebras

Mikael de la Salle

TL;DR

The paper builds a formal bridge between high-rank lattice von Neumann algebra properties and geometric Kakeya phenomena via a rank-0 reduction framework and spherical harmonic analysis. It proves that if the noncommutative $L_p$ space $L^p(\mathcal{L} \mathrm{SL}_{2d-1}(\mathbf{Z}))$ has the operator space approximation property for some $p\neq 2$, then the Kakeya conjecture holds in dimension $d$, with quantitative obstructions linked to tubes and Besicovich sets. The approach develops both qualitative and quantitative regularity results for radial Fourier and Schur multipliers, showing that bounded radial multipliers force the primitive to be smooth in Zygmund sense and that directional square-function estimates govern possible discontinuities; spherical analogues and a refined rank-0 reduction connect these harmonic analysis conclusions back to OAP failures in high-rank lattice algebras. The work synthesizes noncommutative $L_p$ theory, Besov/Littlewood-Paley techniques, and transference on spheres to establish rigidity phenomena, offering a new mechanistic route from operator-algebra structure to geometric incidence conjectures. The results thus provide a formal mechanism to translate analytic regularity constraints into Kakeya-type volume/ dimension consequences in dimension $d$.

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.

Kakeya conjecture and High-Rank Lattice von Neumann algebras

TL;DR

The paper builds a formal bridge between high-rank lattice von Neumann algebra properties and geometric Kakeya phenomena via a rank-0 reduction framework and spherical harmonic analysis. It proves that if the noncommutative space has the operator space approximation property for some , then the Kakeya conjecture holds in dimension , with quantitative obstructions linked to tubes and Besicovich sets. The approach develops both qualitative and quantitative regularity results for radial Fourier and Schur multipliers, showing that bounded radial multipliers force the primitive to be smooth in Zygmund sense and that directional square-function estimates govern possible discontinuities; spherical analogues and a refined rank-0 reduction connect these harmonic analysis conclusions back to OAP failures in high-rank lattice algebras. The work synthesizes noncommutative theory, Besov/Littlewood-Paley techniques, and transference on spheres to establish rigidity phenomena, offering a new mechanistic route from operator-algebra structure to geometric incidence conjectures. The results thus provide a formal mechanism to translate analytic regularity constraints into Kakeya-type volume/ dimension consequences in dimension .

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 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.
Paper Structure (21 sections, 33 theorems, 188 equations)

This paper contains 21 sections, 33 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Rank $0$ reduction
  3. Continuity for the symbol of radial Schur and Fourier multipliers
  4. Preliminaries
  5. Schur multipliers
  6. Non-commutative $L_p$ spaces and completely bounded maps
  7. The Besicovich construction and comparison with usual forms of the Kakeya conjecture
  8. Besov spaces and Littlewood-Paley decomposition
  9. Partitions of unity arguments
  10. From approximation properties of von Neumann algebras to regularity of Schur multipliers
  11. Qualitative regularity of radial Fourier multipliers
  12. Reduction to square function estimates
  13. Investigating directional square inequalities
  14. The conclusion
  15. Quantitative regularity of radial Fourier multipliers
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let $d \geq 2$. If $L^p(\mathcal{L} \mathrm{SL}_{2d-1}(\mathbf{Z}))$ has the operator space approximation property for some $p \neq 2$, then Conjecture conj:Kakeya holds in dimension $d$.

Theorems & Definitions (67)

  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.7
  • Theorem 1.9
  • Corollary 1.10
  • Theorem 1.11
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 57 more