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The three-dimensional Kakeya conjecture, after Wang and Zahl

There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set con…

There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set conjecture! Here we shall be a little vague as to what means here, but roughly one should think of this as “up to factors of the form for any “; in particular this notation can absorb any logarithmic losses that might arise for instance from a dyadic pigeonholing argument. (Actually, to make the induction work, one has to consider a more general family of tubes than these, satisfying some standard “Wolff axioms” instead of the direction separation hypothesis; but we will gloss over this issue for now.)

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