<p>It was <a href="http://www.cs.princeton.edu/~zdvir/papers/Dvir09.pdf">proved by Dvir</a> that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. </p>

<p>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]. </p>

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

<p>Moreover, what is the state of the art for $n &gt; 2$? 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)</p>

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

<p>[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. </p>

<p>[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. </p>

CC BY-SA 3.0
2015-03-31T03:36:21.460Z
Have you seen the papers by Shubhangi Saraf and Swastik Kopparty some with and some without their advisor Madhusudan ?
Student
CC BY-SA 3.0
2015-03-31T04:16:37.380Z
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.
Anurag
<co.combinatorics><coding-theory><extractors><finite-fields>
Applications of small Kakeya sets over finite fields
Anurag
# Applications of small Kakeya sets over finite fields

(Asked by: username_0 on 2015-03-30)

It was [proved by Dvir](http://www.cs.princeton.edu/~zdvir/papers/Dvir09.pdf) 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.

**Tags:** co.combinatorics, coding-theory, extractors, finite-fields

> **Comments:**
> **username_1**: Have you seen the papers by Shubhangi Saraf and Swastik Kopparty some with and some without their advisor Madhusudan ?  
> **username_0**: 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.  

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## 0 answers

