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 conjecture! This conjecture asserts that a Kakeya set – a subset of that contains a unit line segment in every direction, must have Minkowski and Hausdorff dimension equal to three. (There is also a stronger “maximal function” version of this conjecture that remains open at present, although the methods of this paper will give some non-trivial bounds on this maximal function.) It is common to discretize this conjecture in terms of small scale . Roughly speaking, the conjecture then asserts that if one has a family of tubes of cardinality , and pointing in a -separated set of directions, then the union of these tubes should have volume . 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 ... Read full article.