Stickiness, graininess, planiness, and a sum-product approach to the Kakeya problem

Terence Tao
University of California, Los Angeles (UCLA)

We describe a possible approach to the Kakeya conjecture in three-dimensional Euclidean space that relies on stickiness, graininess, and planiness axioms of a putative counterexample to the Kakeya problem to reduce matters to a sum-product estimate, which (in principle at least) is a consequence of a sum-product estimate of Bourgain. The required stickiness, graininess, and planiness axioms are known to hold if the optimal counterexample has dimension 5/2 (and one is in the Minkowski setting), but in other settings these axioms are only partially verified, so this does not yet constitute a full solution to the Kakeya conjecture even in principle. This is joint work with Nets Katz.

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