Terence Tao on Nostr: Two of the most notorious open problems in harmonic analysis are the restriction ...

Two of the most notorious open problems in harmonic analysis are the restriction conjecture, which concerns how the Fourier transform behaves when restricted to surfaces such as the sphere, and the Kakeya conjecture, which concerns how lines pointing in different directions in space can overlap each other. It has long been known that the restriction conjecture implies the Kakeya conjecture, as any configuration of highly overlapping lines can lead to a configuration of highly interfering "wave packets" that can create unusually large Fourier transforms on surfaces. But getting the converse implication has proven frustratingly difficult: the oscillatory nature of the Fourier transform does not seem to be directly captured by the content of the Kakeya conjecture. While some partial implications do exist, they are inefficient; in particular, a full resolution of the Kakeya conjecture today would not immediately lead to a resolution of the restriction conjecture.

In a recent paper https://arxiv.org/abs/2411.08871, Hong Wang and Shukun Wu have finally obtained an efficient implication of the desired form: they have located a plausible "Kakeya-like" conjecture (a version of the Furstenberg set conjecture) which would imply the full restriction conejcture! As one byproduct of their analysis, they obtain a new "world record" for the restriction exponents in three dimensions, that match the numerology for the benchmark result of Wolff on the Kakeya conjecture in this dimension.