Submit progress

on The Arithmetic Kakeya Conjecture

The statement

A Kakeya set in $\mathbb{R}^d$ is a bounded set which contains a unit line segment in every direction. Such sets can have measure zero, but the Kakeya conjecture posits that they always have (Minkowski and Hausdorff) dimension $d$ . It remains open for $d \ge 4$ .

A related conjecture is the arithmetic Kakeya conjecture, which can be formulated as stating that a property, $AK(\cdot)$ holds of a real number $\alpha$ . It is trivial that $AK(2)$ holds. The arithmetic Kakeya conjecture is that $AK(1)$ holds, and this is known to imply the Kakeya conjecture. The current state-of-the-art is $AK(\gamma)$ where $\gamma \approx 1.675$ is the largest root of $x^3 − 4x + 2$ .

Due to work of Katz and Tao, it is possible to establish $AK(\alpha)$ for particular values of $\alpha$ by constructing certain finite, combinatorial objects. This method is responsible for the current state-of-the-art result. The goal of this problem is to improve on that, i.e. to exhibit such an object that establishes $AK(\alpha)$ for $\alpha < \gamma$ . There are limits known to this method: it cannot work for $\alpha < \frac{3}{2}$ . Thus, this problem cannot establish the full Kakeya conjecture.

See the write-up for full definitions and additional background.

This machinery also facilitates posing a problem asking for an improved upper bound on the sum-difference constant. The best-known bound is 11/6 and any improvement is probably at least moderately interesting.