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)
Nearby in the stack
, which improves on previously known constructions, and which is conjecturally optimal. We further derive consequences for the
Lp
-boundedness of radial Fourier multipliers in terms of Besov spaces with logarithmic smoothness.