Kakeya type problems

A Kakeya set in d-dimensional Euclidean space is a (compact) set which contains a unit line segment in all directions. The Kakeya conjecture is that the Hausdorff dimension of such a Kakeya set must be d. This conjecture is open for all d>3. A rough heuristic for this conjecture is that a line segment is 1-dimensional and the set of directions is of dimension d-1 and, since the set of directions is suitably transverse, the dimensions should add. This type of heuristic applies much more generally. I have worked on the Kakeya problem in various contexts (often avoiding the problem itself!). Some of my work on Kakeya type problems includes variants such as the Furstenberg set problem, (d,k)-sets, and the Kakeya problem in vector spaces over finite fields:

► Hausdorff dimension of restricted Kakeya sets (with L. Yang), arXiv
        submitted.

► Fourier analytic properties of Kakeya sets in finite fields, arXiv
        submitted.

► On variants of the Furstenberg set problem, arXiv
        submitted.

► Fourier decay of product measures, arXiv
        submitted.

► On the Fourier dimension of (d,k)-sets and Kakeya sets with restricted directions (with T. Harris and N. Kroon), arXiv
        Mathematische Zeitschrift, 301, (2022), 2497-2508.

► Some results in support of the Kakeya conjecture (with E. J. Olson & J. C. Robinson), arXiv
        Real Analysis Exchange, 42, (2017), 253-268.

The Kakeya problem is related to Fourier restriction, another problem I have worked a bit on:

► L² restriction estimates from the Fourier spectrum (with M. Carnovale and A. E. de Orellana), arXiv
        submitted.

► An improved L² restriction theorem in finite fields (with F. Rakhmonov), arXiv
        submitted.

Two Kakeya sets in the plane. Are you convinced? Try to find a direction which is not witnessed!