# Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$.

For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ as given by Blokhuis and Mazzocca in [1], where they also classified the sets attaining this lower bound. For $q$ even the bound is $q(q+1)/2$ and all Kakeya sets attaining this bound are known. Some more examples of "small" Kakeya sets are given in [2] and [3].

My question is, what are some possible applications of constructing these small Kakeya sets in $\mathbb{F}_q^2$?

Moreover, what is the state of the art for $n > 2$? What are the best known bounds and the examples that achieve those bounds? Would it be worthwhile to construct explicit examples attaining the bounds there? (which is of course subjective)

[1] A. Blokhuis and F. Mazzocca. The finite field kakeya problem. In Building Bridges, pages 205–218, 2008. http://link.springer.com/chapter/10.1007%2F978-3-540-85221-6_6

[2] A. Blokhuis, M. De Boeck, F. Mazzocca, L. Storme. The Kakeya problem: a gap in the spectrum and classification of the smallest examples. Des. Codes Cryptogr., 72 (1) (2014), 21–31.

[3] J. M. Dover and K. E. Mellinger. Small Kakeya sets in non-prime order planes. European J. of Combin. Volume 47 (2015), pages 95–102.

## 1 Answer

I became interested in Kakeya sets because they have the interesting property that a Kakeya set in a projective plane cannot be a subset of a blocking set, and with the exception of the full plane with two lines removed, Kakeya sets are the only sets with this property.

As far as state of the art in higher dimensions, not much seems known. The article http://onlinelibrary.wiley.com/doi/10.1002/jcd.21507/full, by Maarten De Boeck makes some strides in 3-space, and is an interesting paper.

UPDATE: Shouldn't post so late in the day. I should have said that the affine part of a Kakeya set has the property of not being contained in a blocking set, and that they are almost the only sets not containing a line with that property.