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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.

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  • $\begingroup$ Have you seen the papers by Shubhangi Saraf and Swastik Kopparty some with and some without their advisor Madhusudan ? $\endgroup$ – Anirbit Mar 31 '15 at 3:36
  • $\begingroup$ I am reading one of their papers right now but I don't see any direct application of the explicit examples of small Kakeya sets, as constructed in the papers I have referred to in my question. Perhaps someone with a better understanding of this would be able to shed some light on that. $\endgroup$ – Anurag Mar 31 '15 at 4:16

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