A construction of Kakeya Sets in Arbitrary Dimension

2026/07/16 by Rafael J. Fernández-Delgado, Mikael de la Salle
#math.CA #math.FA

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Abstract

We construct Kakeya sets in arbitrary dimension d≥ 2, generalizing the classical Perron tree construction beyond dimension 2. The Kakeya sets we construct have δ-neighbourhood of volume at most C|logδ|^-(d-1), which improves on previously known constructions, and which is conjecturally optimal. We further derive consequences for the L^p-boundedness of radial Fourier multipliers in terms of Besov spaces with logarithmic smoothness.

Citations

• Spectral Dimension Flow, Localized Non-Orientable Horizon Defects, and Emergent Time in Black Holes: A Geometry-Driven Effective Framework
• The Multiplier Problem for the Ball

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• Fourier analytic properties of Kakeya sets in finite fields
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