A construction of Kakeya Sets in Arbitrary Dimension
arXiv:2607.14824
summary
The paper presents a new method for building Kakeya sets in any dimension, achieving very small volume for their δ‑neighbourhoods and exploring implications for Fourier multiplier bounds.
Abstract
We construct Kakeya sets in arbitrary dimension , generalizing the classical Perron tree construction beyond dimension . The Kakeya sets we construct have -neighbourhood of volume at most , which improves on previously known constructions, and which is conjecturally optimal. We further derive consequences for the -boundedness of radial Fourier multipliers in terms of Besov spaces with logarithmic smoothness.
Corrected typos. Associated GitHub: https://github.com/rafaeljfernandezd/A-CONSTRUCTION-OF-KAKEYA-SETS-IN-ARBITRARY-DIMENSION
Topics & keywords
#geometric measure theory#harmonic analysis#kakeya sets#fourier analysis#Besov spacesKakeya setPerron treeδ-neighbourianlogarithmic smoothnessradial Fourier multipliers