In 1917, Besicovitch constructed a set of Lebesgue measure zero in Rⁿ that `paradoxically’ contains a line segment in every direction. The Kakeya conjecture claims that one cannot further compress such a union of line segments into a `smaller’ set. More precisely, the conjecture is that a measurable set in Rⁿ that contains a line segment in every direction, must have maximal Hausdorff dimension n.
A classical argument shows that if the Fourier extension conjecture holds, namely that the Fourier transform is bounded from the L^p space on the sphere to the L^p space in Rⁿ for p>2n/n-1, then the Kakeya conjecture holds. This argument uses Khintchine’s inequality, which potentially weakens the conjecture that is needed. This leads to the formulation of a probabilistic Fourier extension conjecture, which averages over certain `wavelet’ multipliers by ±1. A part of the proof of this probabilistic conjecture is sufficient for proving the Kakeya conjecture, when combined with a classical argument and a nondegeneracy property of the Fourier transform of smooth Alpert wavelets.
The probabilistic Fourier extension conjecture is proved by constructing smooth Alpert wavelets and applying classical techniques, in particular stationary phase and interpolation.